FUNDAMENTAL  IDEAS 


MECHANICS 


AND 


EXPERIMENTAL  DATA. 


A.   MORIN. 


REVISED,  TRANSLATED,  AND  REDUCED  TO  ENGLISH  UNITS  OF  MEASURE 

BY 

JOSEPH  BENNETT, 


NEW    YORK: 
D.APPLETON    AND    COMPANY, 

346  &   348  BROADWAY. 

LONDON:    16    LITTLE    BRITAIN. 

1860. 


ENTERED,  according  to  Act  of  Congress,  in  the  year  I860,  by 

D.  APPLETON  &  CO., 
In  the  Clerk's  Office  of  the  District  Court  of  the  Southern  District  of  New  York. 


DEDICATORY  PREFACE. 

To  GENERAL  J.  G.  SWIFT: 

In  the  summer  of  1857, 1  was  engaged  upon  a  survey  of 
Lakes  Squam  and  Newfound,  with  a  view  of  making  some 
alterations  in  the  delivery  of  those  great  reservoirs  of  the 
Lowell  water  power ;  on  completing  the  work,  my  employer, 
James  B.  Francis,  made  me  a  present  of  his  only  copy  of 
"  Arthur  Morin's  Legons  de  Mecanique  Pratique ;"  speaking 
of  its  value  in  terms  of  the  highest  commendation,  but 
expressly  stating  that  the  gift  was  not  to  be  considered  as 
involving  the  expectation  of  a  translation,  or  as  anywise  im- 
posing upon  me  an  obligation  to  attempt  it. 

The  temptation  proved  too  strong  to  be  resisted ;  though, 
could  I  have  foreseen  the  labor  required  in  the  reductions,  I 
should  hardly  have  ventured  upon  the  undertaking. 

It  will  be  seen  that  I  have  not,  in  all  cases,  adhered  to  the 
unit  of  the  foot.  In  the  matter  of  the  "  Draught  of  Vehicles  " 
and  "  Resistance  of  Fluids,"  the  near  approach  of  the  yard 
to  the  metre,  and  a  fidelity  of  translation,  seemed  to  call  for 
its  substitution  in  place  of  the  foot.  The  tables  were  not 
merely  transferred  to  our  units,  but  were  calculated  from  the 
data,  as  a  double  check  upon  the  correctness  of  the  original, 
and  its  translation,  and  whatever  errors  may  have  crept  in 
the  latter,  it  is  certain  that  some  grave  errors  of  the  former 
have  been  discovered  and  corrected. 

In  the  calculations,  I  have  been  helped  by  many  friends, 
and  I  appreciate  their  valuable  assistance  in  clearing  the  way- 
through  this  forest  of  figures. 

We  all  acknowledge  the  advantage  of  well  established 
practical  formulae ;  to  them  the  mechanic  and  engineer  must 


IV  DEDICATORY   PREFACE. 

look  for  ready  aid  in  producing  harmonious  combinations  of 
strength  and  dimensions,  ensuring  to  their  mechanical  devices 
and  structures,  agreeable  forms,  convenience,  security,  and 
economy :  it  is  a  matter  of  regret  that  so  little  has  been  done 
in  our  country  towards  establishing  them. 

It  is  not  every  one  like  Brunei  can  congratulate  his 
employers  upon  the  falling  of  a  bridge,  on  the  score  of  its 
preventing  the  erection  of  a  hundred  more  on  the  same  plan. 
With  us,  the  fall  of  one  would  be  "  the  hoisting  of  the  engi- 
neer with  his  own  petard."  The  recent  calamity  at  Lawrence 
(Mass.)  cries  out  in  thunder  tones  against  the  merciless 
destruction  of  life,  and  most  painfully  shows  that  too  much 
care  or  skill  can  not  be  exacted  of  our  constructors.  The 
frequent  record  of  loss  of  life  or  property,  arising  from  a  want 
of  skill  in  those  intrusted  with  the  management  of  our  most 
vital  interests,  has  been  creating  a  wide-felt  disgust  for  the 
too  prevalent  system  of  placing  them  in  hands  who  have  no 
other  claim  but  that  of  political  or  partisan  preference,  and 
its  evil  influences  have  been  operating  upon  a  profession 
which,  in  point  of  attainment  or  utility,  should  stand  second 
to  none  in  the  country. 

It  is  to  be  hoped  that  our  government  may  yet  take  in 
hand  a  matter  that  cannot  well  be  done  at  individual  cost, 
and  thus  institute  a  series  of  experiments,  so  that  for  the 
strains  of  wood,  of  iron,  for  the  properties  of  materials,  and 
general  experimental  results,  there  may  be  found  many  an 
able  native  Barlow,  Fairbairn,  or  Morin,  to  elicit  valuable 
information,  and  supply  the  great  void  existing  in  the  testing 
of  our  own  materials. 

In  dedicating  this  translation  to  one  of  your  great  ability 
and  experience,  I  express  the  hope  that  I  have  done  justice 
to  the  gifted  author,  and  that  in  presenting  the  results  of  his 
ingenious  experiments,  I  may  have  done  the  profession  some 
service. 

Most  respectfully, 

Jos.  BENNETT. 

BROOKLYN,  JAN.  13,  1860. 


CONTENTS. 


PRELIMINARY  IDEAS. 

PAGE 

Extension 1 

Simpson's  formula 2 

Quantity  infinitely  divisible 5 

FORCES  AND  THE  MEASURE  OF  THEIR  WORK. 

Inertia  of  matter. — Definition  offeree 8 

Action  offeree 10 

Measure  offeree. — The  unit  of  measure 11 

Different  names  of  forces. — Constitution  of  bodies 12 

Action  andreaction,  equal  and  opposite 13 

Point  of  application  of  forces. — Effect  and  work  of  forces 14 

Measure  of  work  of  a  constant  force,  where  the  path  described  by  its  point 

of  application  is  in  its  own  direction 15 

Representation  of  this  work  by  the  surface  of  a  triangle. — Measure  of  the 

work  of  a  variable  force 16 

Mean  effort  of  a  variable  force 19 

Mode  of  calculation  in  English  practice. — Case  where  the  arithmetical 
mean  of  variable  values  may  be  taken  for  the  mean  effort. — Applica- 
tions   20 

Idea  of  work  independent  of  time 21 

Denominations  and  unit  of  mechanical  work 22 

Conditions  of  mechanical  work 23 

Horizontal  transportation  of  loads 24 

Case  where  the  force  does  not  act  in  the  direction  of  the  path  described....  25 

The  work  of  gravity  upon  a  body  describing  any  curve 26 

The  crank  and  its  connecting  rod. — Direction  of  effort  in  its  relation  to  the 

path  described 27 

Springs. — Expansion  and  contraction 28 

Proper  limits  to  variations  of  temperature  to  be  used 29 


VI  CONTENTS. 

DYNAMOMETERS. 

PAGE 

Conditions  to  be  fulfilled  by  these  instruments 33 

Rules  for  proportioning  spring-plates 34 

Ratio  between  the  different  proportions 35 

Longitudinal  profile  of  plates. — Disposition  of  plates 36 

Permanent  trace  of  spring  flexures 38 

Motion  of  paper  receiving  the  trace  of  the  style 38 

Quadrature  of  traced  curves 39 

The  Planimeter 41 

Dynamometer,  showing  the  whole  quantity  of  action  for  a  considerable 

interval  of  time  and  space 45 

Indications  of  the  number  of  turns  made  by  small  wheel  dynamometer,  with 

chronometer  motors 48 

Rotating  dynamometer,  with  styles 49 

Transmission  of  the  motion  of  shaft  to  the  band  of  paper 51 

Results  of  experiments  made  with  the  rotating  dynamometer 51 

Rotating  dynamometer  with  counter. 52 

Watt's  gauge,  perfected  by  MacNaught 53 

New  style  indicator 55 

TRANSMISSION  OF  MOTION  BY  FORCES. 

General  remarks  upon  the  laws  of  motion 58 

Vertical  motion  of  heavy  bodies 59 

Successive  fall  of  heavy  bodies 60 

Forces  proportional  to  their  velocities 61 

Measure  of  motive  forces  and  of  inertia 62 

Case  where  the  force  is  constant. — Relation  offerees  to  accelerations 64 

Quantity  of  motion 65 

Equal  forces  acting  during  equal  times 66 

Proof  of  preceding  considerations  by  direct  experiment 69 

Shock  of  two  elastic  bodies 73 

Observations  upon  the  preceding  results , 74 

Quantity  of  motion  imparted  bya  constant  force 75 

Observations  upon  the  use  of  quantity  of  motion 78 

OBSERVATION  OF  THE  LAWS  OF  MOTION. 
Determination  of  the  intensity  of  forces  by  observing  the  laws  of  the  motions 

they  produce 80 

Means  of  determining  the  laws  of  motion. — Colonel  Beaufoy's  and  Eytele- 

wein's  apparatus 81 

New  apparatus 82 

Zinc  plates. — Contrivance  for  tabulating  the  curves , 88 

Description  of  a  chronometric  apparatus  with  cylinder  and  style,  for  observ- 
ing the  laws  of  motion 89 

Discussion  of  results  furnished  bythis  apparatus 91 


CONTENTS.  Vil 

PAGE 

Determination  of  the  velocity 92 

Experimental  demonstration  of  the  principle  of  the  proportionality  of 
forces  to  the  velocities 93 

PRINCIPLE  OF  VIS  VIVA. 

Mechanical  work  developed  byforces,  in  variable  motion 97 

Vis  viva 98 

Effects  of  powder  in  fire-arms. 99 

Application  of  the  principle  of  vis  viva 101 

Relation  between  charges  and  velocities. 102 

Initial  velocities  and  vis  viva  imparted  to  balls  by  different  charges  of 

gunpowder 104: 

Vis  viva  imparted  by  different  powders.  — Mean  efforts 106 

Effects  of  powder  and  pyroxile  on  fire-arms 107 

Consumption  and  restoration  of  work  by  inertia 113 

Rams,  punching-machines,  &c 113 

Work  expended  in  the  shocks  of  two  non-elastic  bodies 114 

Work  due  to  compression,  and  the  return  to  the  primitive  form  in  the 

case  of  elastic  bodies. 116 

Work  lost  in  the  shock  of  bodies  imperfectly  elastic 116 

Masses  in  motion  reservoirs  of  work 117 

Periodical  motion 118 

COMPOSITION  OF  MOTION,  VELOCITIES,  AND  FOECES. 

Composition  and  resolution  of  simultaneous  motions 119 

Simultaneous  motions  in  same  direction 120 

Composition  of  motions  directed  in  any  manner 121 

Variable  motion 124 

Components  at  right  angles 125 

Resultant  of  three  simultaneous  motions  or  velocities  in  space 127 

Resultant  of  any  number  of  simultaneous  motions  or  velocities 128 

Varignon's  theorem  of  moments 129 

Extension  of  these  theorems  to  systems  impressed  with  a  common  motion 

of  translation. — Independence  of  the  simultaneous    action  of  forces 

upon  the  same  point 132 

Forces  acting  in  different  directions 134 

Quantity  of  work  of  a  force  whose  point  of  application  does  not  move  in 

the  same  direction  as  the  force 135 

Application  of  Varignon's  theorem  to  forces. — Resultant  work  of  forces 

equal  to  algebraic  sum  of  its  component  works ...  136 

Forces  acting  in  any  direction 137 

Case  of  the  point  turning  around  a  fixed  axis 138 

Conditions  of  uniform  motion  or  equilibrium,  the  forces  being  in  the  same 

plane. 139 

Forces  acting  in  any  manner  in  space 140 


Ylll  CONTENTS. 

PAGE 

Parallel  forces 142 

Consequence  of  the  composition  of  parallel  forces ; 143 

Point  of  application  of  resultant  of  parallel  forces 143 

Work  of  resultant  of  parallel  forces 148 

Centre  of  parallel  forces 148 

Use  of  moments  in  determining  position  of  resultant 149 

Condition  of  uniform  motion  or  of  equilibrium 149 

The  balance 150 

Proof  of  balances 154 

Double  weighing. — The  steelyard 155 

Steelyard  with  a  fixed  weight  (Peson) 157 

Quinteux's  platform  balance. 159 

Theory  of  the  lever 162 

THE  CENTEE  OF  GEAVITY  AND  EQUILIBEIUM  OF  TENSIONS  IN 
JOINTED  SYSTEMS. 

Application  of  preceding  theorems  to  gravity 165 

Determination  of  the  centre  of  gravity 165 

Geometrical  method. — Triangle 166 

Quadrilaterals. — Polygons .  — Triangular  pyramid 167 

Centre  of  gravity  of  a  body  of  any  form 168 

The  stability  of  equilibrium 169 

Application  of  the  principles  of  composition  and  resolution  of  forces 170 

Equilibrium  of  cords 171 

Equilibrium  of  efforts  transmitted  by  cords  or  rods  meeting  in  the  same 

point 172 

Movable  pulley. — Towers 172 

Funicular  polygon 173 

Weights  acting  upon  the  funicular  polygon 174 

Determination  of  the  tensions  by  a  graphical  construction , 175 

Suspension  bridges 176 

Application....... 180 

COMPOSITION  AND  EQUILIBEIUM  OF  FOECES  APPLIED  TO  A  SOLID  BODY. 

Forces  applied  to  solid  bodies.— Motion  of  translation  of  a  system  of  bodies 

parallel  to  itself. 182 

Case  of  variable  motion ...; 183 

Quantity  of  motion  and  vis  viva  of  a  body 184 

Work  of  gravity  in  compound  systems 185 

A  system  of  forces,  acting  upon  a  solid  body,  may  always  be  reduced  to 
two  equivalent  forces,  applied  to  two  of  its  points,  one  of  which  may 

be  chosen  at  will 186 

Condition  of  uniformity  of  motion  or  of  equilibrium 187 


CONTENTS.  IX 


MOTION  OF  ROTATION. 

PAGE 

Work  and  equilibrium  of  forces,  in  the  motion  of  rotation  around  a  fixed 

axis 189 

General  conditions  of  the  uniformity  of  motion  or  of  equilibrium  of  a  solid 

body,  free  in  space,  and  subjected  to  any  forces 191 

Centrifugal  force — its  measure 193 

•"Work  developed  by  centrifugal  force 195 

Action  of  centrifugal  force  upon  wagons 197 

Action  of  centrifugal  force  in  fly-wheels 198 

Application  to  the  motion  of  water  in  a  vase  turning  round  a  vertical  axis  199 
Surface  of  water  in  the  bucket  of  a  hydraulic  wheel  with  a  horizontal  axle  201 

Regulators  with  centrifugal  force 202 

Distribution  of  a  regulator  with  centrifugal  force 207 

Results  of  observations  upon  the  effect  of  this  regulator 210 

Comparison  of  the  data  of  experiment  with  the  formula. — Modification  of 

the  balls  for  obtaining  a  greater  regularity 212 

Transmission  of  motion  by  the  endless  screw 213 

Indispensable  disposition  in  the  use  of  these  regulators 214: 

Modification  of  apparatus. — Other  regulators. — Variable  motion  around 

an  axis 215 

Observations  upon  the  moment  of  inertia 217 

Principle  of  vis  viva  in  the  motion  of  rotation  about  an  axis 219 

Theory  of  the  pendulum 221 

Time  of  oscillations  of  a  pendulum  with  small  vibrations 224 

Compound  pendulum 226 

Length  of  the  simple  pendulum  which  makes  its  oscillations  in  the  same 

time  as  the  compound  pendulum 228 

Moment  of  inertia  of  a  compound  pendulum 228 

Centre  of  gravity  of  compound  pendulums 230 

Centre  of  percussion 231 

Theory  of  the  ballistic  pendulum 234 

APPLICATION  OF  THE  PRINCIPLE  OF  VIS  VIVA  TO  MACHINES. 

Application  of  "  vis  viva  "  to  machines 240 

Maximum  effect  of  machines. — Work  of  powers  and  of  useful  resistances  242 

Work  of  passive  resistances. — Pieces  with  alternating  motion 243 

Influence  of  vis  viva  acquired  at  each  period 244 

Periodical  motion 245 

Advantages  and  conditions  of  uniform  motion. 245 

Means  of  diminishing  variable  motion 246 

Observations  upon  the  starting  of  machines,  and  the  variations  in  velocity 

which  then  take  place 246 

Perpetual  motion. — Periodical  motion 248 

Limitations  of  the  deviation  of  velocity. — Theory  of  fly-wheels 249 


X  CONTENTS. 

PACK 

High-pressure  steam-engines 253 

Fly-wheels  for  expansion  engines,  and  for  forge-hammers 254 

German  hammers  geared 255 

Geared  tilt-hammers. — Necessity  of  using  fly-wheels  when  there  are- 
shocks 256 

Proportions  of  fly-wheels  for  powder-mills  with  twenty  stamps. — Rolling- 
mill  for  great  plates 257 

Use  of  fly- wheels 258 

FBICTIOK 

Ancient  experiments 260 

Experiments  at  Metz  and  description  of  apparatus 262 

Graphic  results  of  experiments 223 

Formulae  for  calculating  results  of  experiments 264 

Relations  between  the  tensions  of  the  cord  and  friction  of  the  sled 268 

Friction  of  oak  upon  oak,  without  unguents .- 270 

Friction  of  elm  upon  oak,  without  unguents 271 

Friction  of  soft  limestone  upon  soft  limestone,  without  unguent 272 

Friction  of  strong  leather,  placed  flatwise  upon  cast-iron 273 

Friction  of  brass  upon  oak,  without  unguent 274 

Friction  of  cast-iron  upon  cast-iron 275 

Experiments  upon  friction  at  starting,  and  results  of  experiments 276 

Friction  of  oak  upon  oak,  the  fibres  of  the  sliding  pieces  being  perpendicu- 
lar to  those  of  sleeper k  277 

Friction  of  oak  on  oak,  the  sliding  pieces  having  their  fibres  vertical, 
while  those  of  the  fixed  pieces  are  horizontal  and  parallel  to  the  direc- 
tion of  motion 278 

Friction  of  limestone  upon  limestone,  when  the  surfaces  have  been  some 

time  in  contact 279 

Friction  of  limestone  upon  limestone,  the  surfaces  having  been  in  contact 

with  a  bed  of  mortar 280 

Expulsion  of  unguents  and  influence  of  vibrations  upon  friction  at  starting  281 

Influence  of  unguents. — Adhesion  of  mortar 282 

Experiments  upon  friction  during  a  shock 283 

Apparatus  employed  in  the  experiments 284 

General  circumstances  of  the  experiments 285 

Formula  for  calculating  the  results  of  experiments 286 

Acceleration  of  motion  of  sledge  during  fall  of  shell  may  be  neglected....  287 

Results  of  experiments 289 

Friction  of  cast-iron  upon  cast-iron,  with  an  unguent  of  lard,  during  the 

shock 290,  291 

Transmission  of  motion  by  belts. — Slipping  of  belts  upon  cylinders 292 

Slipping  of  cords  and  belts  upon  wooden  drums  and  cast-iron  pulleys 294 

Friction  of  belts  upon  wood  drums 296 


CONTENTS.  XI 

PAGE 

Friction  of  belts  of  curried  leather  upon  cast-iron  pulleys 29  7 

Conclusions 298 

Variation  of  the  tension  of  belts  in  transmitting  motion 299 

Remarks  upon  preceding  results 306 

Friction.of  journals 307 

Friction  of  cast-iron  journals  upon  cast-iron  bearings 309 

Advantage  of  granulated  metals 310 

Light  mechanisms. — Use  of  experiments 311 

Friction  of  plane  surfaces  which  have  been  some  time  in  contact 312 

Friction  of  plane  surfaces  in  motion  upon  each  other 313 

Friction  of  journals  in  motion  upon  their  pillows. 314 

Application  to  gates 315 

Application  to  saw-frames 316 

Application  to  journals 317 

Axles  of  wagons 320 

RIGIDITY  OF  COEDS. 

Rigidity  of  cords. — Experiments  of  Coulomb  with  apparatus  of  Amontons  321 

Results  of  Coulomb's  experiments 323 

General  expression  of  resistance  to  rolling 325 

Other  experiments  of  Coulomb 328 

Rigidity  of  cords  with  movable  rollers  upon  a  horizontal  plane 329 

Extension  of  Coulomb's  experiments  to  those  of  different  diameters 330 

Rigidity  of  cords  in  function  of  number  of  strands 331 

Remarks  on  cords  that  have  been  used 332 

Tarred  cords 333 

Rigidity  of  cords  of  different  diameters  rolling  upon  a  drum  one  foot  in 

diameter 334 

Moistened  cords. — Use  of  preceding  tables 336 

DRAUGHT  OF  VEHICLES. 

Draught -of  vehicles 338 

Experiments  on  oak  rollers,  rolling  upon  poplar. — Vehicles  moving  upon 

common  roads 341 

Ratio  of  the  draught  to  the  load 343 

Influence  of  the  pressure 345 

Experiments  upon  the  influence  of  pressure  upon  the  draught  of  vehicles  346 

Influence  of  the  diameter  of  the  wheels 347 

Experiments  upon  the  influence  of  the  diameter  of  wheels  upon  the  resist- 
ance to  the  draught  of  vehicles 348 

Influence  of  the  width  of  the  rims 349 

Influence  of  the  velocity. 350 

Approximate  expression  for  the  increase  of  resistance  with  the  velocity...  351 


Xll  CONTENTS. 

PAGE 

Practical  consequences  of  these  experiments. — Comparison  of  paved  and 

metalled  roads , 354 

Influence  of  the  inclination  of  the  traces 356 

Application  of  the  general  results  of  experiments 359 

Draught  and  load  of  carriages  for  different  soils  and  vehicles 360,  361 

Consequences  relative  to  the  construction  of  vehicles 364 

Destructive  effects  of  vehicles  upon  roads 365 

Influence  of  great  diameters  of  wheels 366 

Direct  experiments  upon  the  destructive  effects  of  wagons  upon  roads 367 

Influence  of  the  width  of  tires. 367 

Experiments  with  the  same  carriages  under  equal  loads 369 

Experiments  made  upon  the  influence  of  the  diameter  of  wheels,  in  their 

destructive  effects  upon  roads ^ 369 

.Influence  of  velocity  upon  the  destructive  effects 370 

Comparative  experiments  upon  the  wear  produced  by  carriages,  carts, 

and  wagons  without  springs 371 

Experiments  to  determine  the  loads  of  equal  wear  and  tear 372 

EESISTANCE  OF  FLUIDS. 

Eesistance  of  fluids 373 

Work  developed  per  second  by  the  resistance  of  a  medium. — Equivalent 

expressions  of  the  resistance 376 

The  body  at  rest  in  a  fluid  in  motion 377 

Experiments  upon  the   resistance  of  water  to  the  motion  of  variously 

formed  bodies ,.  377 

Mode  of  observation 378 

Observations  upon  the  results 379 

Influence  of  the  acuteness  of  the  angles  of  cones  upon  the  resistance 380 

Resistance  of  water  to  the  motion  of  projectiles 381 

Resistance  of  water  to  the  motion  of  floating  bodies. — Influence  of  the 

form  of  floating  bodies 382 

Flat-bottomed  boats  with  raised  fronts 383 

Velocity  of  waves 385 

Experiments  upon  the  velocity  of  solitary  waves  produced  by  boats 388 

Results  of  experiments  upon  the  resistance  of  boats  to  towing. — Fast  boats  389 

Consequences  of  the  experiments 393 

Comparison  of  the  resistance  to  towing  of  mail-boats,  when  the  wave  is 

spread  along  the  sides,  and  when  it  is  towards  the  bow 394 

Work  developed  by  horses  in  hauling  fast  boats 395 

Days  work  developed  by  horses  in  different  modes  of  transportation 397 

Observation  upon  the  daily  work  of  horses 398 

Resistance  of  water  to  the  motion  of  wheels  with  plane  paddles 398 

Causes  which  alter  the  law  of  resistance 401 

Proper  distance  of  paddles  apart. — Value  of  the  second  term  of  the  resist- 
ance K,..  ..  403 


CONTENTS. 

PAGE 

Influence  of  the  presence  of  a  boat  near  the  wheels 405 

Application  to  the  wheels  of  steamboats 406 

Resistance  of  air. — Results  of  experiments 408 

Thibault's  experiments  upon  bodies  inmotion  in  air 410 

Remarks  upon  regulators  and  wind-mills. — Experiments  upon  different 

formed  surfaces 413 

Influence  of  the  inclination  of  the  wings 414 

Approximation  of  surfaces  exposed  to  the  resistance  of  air. — Influence  of 
the  form  of  surfaces. — Resistance  of  air  to  the  motion  of  spherical 

bodies 415 

Experiments  atMetz  upon  bodies  moving  in  air 417 

Mode  of  reckoning  the  effects  of  acceleration 419 

Proof  of  the  exactness  of  the  formula i 421 

Influence  of  the  extent  of  surfaces 422 

Experiments  upon  parachutes 424 

Case  where  the  parachute  presents  its  convexity  to  the  air,  and  where  its 

motion  was  accelerated 425 

Resistance  to  the  motion  of  inclined  planes  in  air. 426 

General  conclusions  from  the  experiments  at  Metz 427 

Effort  exerted  by  the  wind  upon  immovable  surfaces  exposed  to  its  direc- 
tion   428 

Observation  upon  the  velocity  of  the  wind 429 

Means  of  measuring  the  velocity  of  air 430 

Anemometer  of  M.  Combes 431 

Remarks  upon  the  use  of  the  instrument 432 

New  anemometer 433 

Testing  of  the  instrument 435 

Remarks  upon  the  test  of  the  instrument,  and  its  extension  to  great  veloci- 
ties   436 

M.  Thibault's  experiments  upon  the  effort  of  wind  upon  immovable  sur- 
faces, exposed  to  its  action,  perpendicular  to  its  direction 439 

Accordance  of  these  results  with  those  of  Professor  Rouse,  cited   by 

Smeaton. 440 

Influence  of  the  curvature  of  surfaces,  and  of  their  inclination  to  the  wind  441 
Difficulties  in  the  directing  of  balloons 442 


\ 


FUNDAMENTAL  IDEAS  OF  MECHANICS 


AND 


EXPERIMENTAL   DATA. 


PEELIMIJSTAEY  IDEAS. 

1.  Extension. — Extension  has  three  dimensions  :  — 
length,  breadth,  and  depth.  In  its  measurement  we  call 
that  a  line,  or  linear  dimension,  which  has  length  without 
breadth ;  that  a  surface  which  has  length  and  breadth ; 
and  that  a  solid,  or  volume,  which  combines  the  three  di- 
mensions. The  measurement  of  extension  constitutes  the 
science  of  Geometry,  the  use  of  which  in  this  treatise 
will  be  its  application  to  Mechanics. 

Lengths  are  measured  by  a  comparison  with  a  conven- 
tional unit  adopted  in  any  country,  which  with  us  is  the 
foot,  subdivided  into  tenths,  hundredths,  &c.  To  appre- 
ciate any  thing  smaller  than  hundredths,  we  make  use  of 
the  vernier  and  other  contrivances,  such  as  micrometer 
screws,  compensators,  &c.,  whose  description  is  within 
the  province  of  Industrial  Geometry. 

Surfaces  are  measured  by  the  rules  of  Geometry,  and 
are  expressed  in  square  feet.  But  it  is  often  the  case 
that  they  are  bounded  by  lines  and  contours,  not  con- 
forming to  any  known  geometrical  law,  when  we  must 
have  recourse  to  approximate  modes  of  quadrature,  or 
some  mechanical  means.  The  use  of  these  being  a  matter 
of  daily  occurrence,  in  tabular  abstracts,  and  in  the  dis- 


PEELIMINAEY   IDEAS. 


cussion  of  experimental  results,  to  provide  for  any  future 
reference  to  them,  we  will  speak  of  them  somewhat  mi- 
nutely. 

One  of  the  most  sim- 
pie  and  exact  methods  for 
determining  approximate- 
ly  by  calculation,  a  sur- 
face  bounded  by  curved 
lines,  or  partly  composed  of  curved  and  straight  lines,  is 
the  following :  Draw  across  the  surface  a  line  AB,  and 
divide  the  distance  between  the  points  of  its  intersection 
with  the  contour,  into  an  even  number  of  equal  parts, 
numbered  1,  2,  3,  4,  ....  7,  8,  9,  for  example.  At  the 
points  of  division  raise  perpendiculars  to  the  line  AB, 
(called  the  axis  of  abscissa,)  giving  II",  2'2",  3'3"  .... 
8'8",  9'9"  for  the  lengths  of  ordinates.  This  done,  the 
surface  S,  terminated  by  the  curved  line,  will  have  the 

value  nearly  S=|(l,  2)|"l/l//  +  9/9//+4(2/2//+4:/4//+  .  .  . 

8'8")+2(3'3"+5'5"  .  .  .  7'7")1  that  is  to  say,  the  third  of 

the  space  between  two  consecutive,  equidistant  ordinates, 
multiplied  ~by  the  sum  of  the  extreme  ordinates,  plus  four 
times  the  sum  of  ordinates  of  an  even  order,  plus  twice  the 
sum  of  the  ordinates  of  an  uneven  order. 

M.  Poncelet  gives  the  following  demonstration  of  this 
rule,  page  187  of  "  L'introduction  a  la  mecanique  indns- 
trielle."  (Second  Edition.) 

2.  Demonstration  of  Simpson's  formula. — The  area 
to  be  measured  being  limited  by  the  contour  line 


PRELIMINARY   IDEAS. 


3 


.  .  .  g'g  .  .  .  ba,  if  we  divide  the  line  ag  into  six 
equal  parts,  we  shall  have  at  once  a  first  approximation 
by  taking  the  sum  of  the  right  lined  trapeziums 
',  &c.,  which  gives 


,  &c., 


which  is  equal  to 


This  method  is  usually  followed.  But  it  is  manifest  that 
for  curves  always  concave  towards  the  Jine  ag  of  abscissa, 
this  formula  will  give  too  small  a  result  ;  and  otherwise 
it  will  give  too  much  for  curves  convex  toward  the  line 
ag  :  so  that  the  only  approximate  compensation,  will  be 
for  the  case  of  curves  alternately  concave  and  convex. 

But  if  we  consider  the  space 
between  two  odd  consecutive 
ordinates  ccf  and  ee\  and  di- 
vide ce  into  three  equal  parts, 
cm=mn=ne,  we  have  imme- 
diately a  nearer  approximation 
to  the  mixtilinear  area  cc'd'e'e  ; 
by  substituting  the  three  right 
lined  trapeziums  cc'm'm,  mm'n'n^  nn'e'e,  in  place  of  the 
two  trapeziums  cc'd'd  and  dd'e'e  :  the  sum  of  the  area  of 
these  three  trapeziums  is 

- 

o 

since  cm=mn=ne=-ab. 

o 

In  drawing  the  line  m'n1  which  meets  dd!  in  0,  we  have 
<fo=-(mm'+nri)  ;  whence 


1 

; 

j 

i 

i 
j 

I 

PRELIMINARY   IDEAS. 


The  total  area  of  these  three  trapeziums  has  then  for 
its  value, 


!Now  when  the  curve  is  concave  towards  the  axis  of 
abscissa,  this  area  is  smaller  than  the  curvilinear  area 
to  be  measured  ;  and  if  we  substitute  for  do,  the  ordinate 
dd'y  a  little  greater,  and  which  is  given,  we  establish  an 
approximate  compensation. 

The  inverse  occurring,  in  case  the  curve  is  convex 
to  the  axis  of  abscissa,  we  have  an  analogous  com- 
pensation. Then  we  shall  obtain  a  value  more  nearly 
approximating  to  the  curvilinear  area  cdd'e'e  by  the 
expression 


o 
We  have  also  for  the  area  aa'c'c, 

\db(aa'-\-£bV+ccf). 

o 

Then,  taking  the  sum  of  all  these  partial  areas,  we  shall 
have  for  an  approximate  value  of  the  whole  surface 


which  is  Simpson's  formula. 

Should  any  of  the  ordinates  become  zero,  it  will  not 
prevent  the  use  of  the  formula. 

We  should  select  the  line  AB  of  abscissae,  so  that  the 
ordinates  may  not  intersect  the  curve  under  too  small 
angles,  so  as  to  leave  any  uncertainty  as  to  the  point  of 
intersection. 

The  divisions  should  be  increased  in  number  accord- 
ing to  the  salience  or  undulations  of  the  curve,  and  the 
desired  closeness  of  the  approximation. 

When  the  surface  to  be  squared  is  limited  beforehand 
to  a  line  of  abscissae  AB,  and  to  two  extreme  ordinates, 
we  use  the  same  process. 


PEELIMINAEY   IDEAS.  5 

This  method,  called  after  the  name  of  its  author,  is 
more  exact  and  approximate  than  that  of  taking  the  sum 
of  the  inscribed  trapeziums.  We  shall  give  numerous 
applications  of  it. 

The  cubature  of  solids  of  irregular  excavations  and 
embankments,  the  displacement  of  vessels,  afford  frequent 
opportunities  for  its  use. 

In  dealing  with  solids  bounded  by  irregular  curved 
surfaces,  the  laws  of  which  are  unknown,  we  adopt  a  sim- 
ilar method.  We  take,  for  an  example,  the  displacement 
of  vessels :  we  make,  or  usually  have  at  the  outset,  the 
trace  of  the  transverse  profiles,  or  moulds  of  the  ship, 
made  at  equal  distances  from  stem  to  stern,  and  bounded 
on  its  upper  part  by  the  water  line.  We  begin  by  mak- 
ing a  partial  quadrature  of  each  of  these  profiles  taken  in 
odd  numbers,  containing  consequently  an  even  number 
of  equal  spaces.  We  mark  upon  a  line  of  abscissae  these 
equal  spaces.  At  each  point  of  division  a  perpendicular 
or  ordinate  is  to  be  raised,  which,  at  a  convenient  scale, 
is  made  to  represent  the  surface  of  the  corresponding  pro- 
file. Through  the  extremity  of  all  these  ordinates  a  curve 
is  passed,  and  the  area  comprised  between  the  curve  and 
the  exterior  ordinates  of  the  line  of  abscissae,  calculated 
by  the  formula  of  Simpson,  gives  the  volume  of  the  dis- 
placement of  the  vessel. 

We  resolve,  as  we  shall  see  hereafter,  by  the  method 
of  quadrature,  many  other  questions,  the  calculation  of 
which  would  be  attended  with  great  difficulties. 

3.  Divisibility  of  quantities  into  infinitely  small  ele- 
ments.— Before  proceeding  farther,  it  would  be  well  to 
remark,  that  as  all  quantities  are  susceptible  of  increase 
or  diminution,  so  they  may  be  regarded  as  composed  of 
parts,  the  number  of  whose  elements  will  be  so  much  the 
greater,  as  the  parts  are  less ;  and  as  in  the  smallest  part, 
we  may  conceive  of  a  still  smaller,  we  see  in  reality,  that 
quantities  or  bodies  may  be  regarded  as  composed  of 


6  PKELIMINAKY   IDEAS. 

infinitely  small  elements  ;  or  smaller  than  any  given 
quantity,  whose  reunion  or  sum  produces  a  finite  quantity. 

If  we  consider  the  progressive  increase  in  objects 
which  nature  presents  to  us,  we  may  more  readily  con- 
ceive of  the  gradual  increase  or  diminution  of  quantities 
by  the  continuous  addition  or  subtraction  of  infinitely 
small  quantities. 

Vegetables,  in  their  so  various  and  sometimes  so  rapid 
growth,  do  not  shoot  forth  save  by  insensible  degrees,  by 
infinitely  small  developments,  which  added  each  to  the 
other  during  the  month,  the  year,  form  the  growth  of  that 
period. 

Suppose  a  child  between  the  age  of  10  and  12  grows 
0.3ft.  per  year,  or  in 


3600"  x  24A.  x  365d.=  31536000", 
the  growth  per  second  will  be 

-=0.0000000009/*.; 


31536000 

and  as  the  second  can  be  indefinitely  divided,  so  will  it 
be  with  the  increase. 

It  is  thus  the  tramping  of  foot  passengers,  in  years, 
may  wear  away  the  flagstones  of  sidewalks, — that  a  water- 
fall may,  in  centuries,  destroy  the  rocks  of  granite  on 
which  it  falls,  taking  away  and  destroying,  at  each  in- 
stant, infinitely  small  quantities,  which,  each  added  to  the 
other,  will  produce  ultimate  destruction. 

4.  Observations  upon  these  examples. — These  examples 
were  introduced  to.  impress  upon  us  the  fact  that  all  bod- 
ies increase  and  diminish  gradually,  by  infinitely  small 
elements,  which  added  to  each  other  in  finite  times,  form 
finite  quantities. 

These  ideas  will  be  necessary  in  our  study  of  mechan- 
ical effects,  which  are  accomplished  by  degrees  sometimes 


PRELIMINARY   IDEAS.  7 

slow,  and  sometimes  so  rapid  that  their  duration  cannot 
be  appreciated  by  our  senses  and  means  of  observation, 
but  which  in  no  case  can  ever  be  supposed  as  nothing. 
They  permit  us  to  regard  bodies  as  an  assemblage  of  ma- 
terial points,  small  as  we  may  choose,  but  having  all  the 
properties  of  matter,  such  as  weight,  impenetrability,  &c. 


FOKCES  AND  THE  MEASUEE  OF  THEIR  WORK. 

5.  Inertia  of  Matter. — All  ~bodies  maintain  the  state  of 
rest,  or  of  uniform  motion  in  which  they  are  found,  pro- 
vided  no  foreign  cause  ly  its  action  constrains  them  to 
change  that  state.    (Newton's  first  law  of  motion.) 

From  this  fundamental  law  it  follows,  as  we  have  al- 
ready seen,  that  in  variable  motion,  if  the  cause  producing 
the  variation  ceases  its  action,  the  motion  becomes  uni- 
form ;  and  that  in  curvilinear  motion,  if  the  cause  which 
compels  the  body  at  each  instant  to  change  its  direction 
ceases  to  act,  the  motion  will  preserve  the  direction  of  the 
last  described  curvilinear  element,  and  consequently  will 
become  tangential. 

6.  Definition  of  forces. — Force  is  the  name  given  to 
any  cause  which  produces  or  modifies,  or  tends  to  pro- 
duce or  modify  motion.     Such  are  attraction,  gravity,  the 
action  of  animals,  of  water,  of  air,  of  steam,  the  resistance 
of  air,  friction,  &c. 

Since  some  outward  action  is  always  necessary  to 
change  the  state  of  motion  of  a  body,  it  is  apparent  that 
the  body  must  oppose  a  certain  resistance,  arising  from  its 
inertia  ;  or,  as  Newton  defines  it,  "  The  force  residing  in 
matter  (the  inseated  force)  is  the  power  of  its  resistance. 

"  A  body  exerts  this  force  whenever  it  changes  its  ac- 
tual state  of  motion,  so  that  we  may  consider  it  under  two 
different  aspects  ;  either  as  resisting,  in  so  far  as  the  body 
is  opposed  to  the  force  which  tends  to  change  its  state, 


FOECES   AND   THE    MEASURE   OF   THEIR   WORK. 


9 


or  as  impulsive,  inasmuch  as  the  body  itself  makes  an 
effort  to  change  the  state  of  the  obstacle  which  resists  it. 
Thus  we  give  to  the  force  residing  in  bodies,  the  very  ex- 
pressive name  of  force  of  inertia"  (Newton,  Principia, 
etc.,  vol.  i.,  page  2.) 

We  may  prove  by 
example,  that  inertia  is 
a  force,  whose  action  is 
manifest  in  every  change 
of  motion.  Thus,  sup- 
pose a  body  AB  placed 
upon  AD,  and  determ- 
ine by  experiment  the 
weight  P  required  to  be 
hung  at  the  end  of  the  thread  CE  (fastened  to  a  point  C, 
and  passing  over  a  pulley),  to  overturn  the  body  AB  ;  it 
is  clear  that  any  cause,  whether  applied  in  front  or  rear* 
which  overturns  the  body,  supposed  to  be  symmetrical, 
will  be  equivalent  to  the  weight  P,  and  will  be  a  force. 

Now  if  we  give  the  plane  AD  an  accelerated  motion, 
we  shall  see,  if  the  acceleration  is  made  with  a  certain 
rapidity,  that  the  body  AB  will  be  upset  in  a  direction 
opposite  to  the  motion.  Its  inertia  will  have  acted  in  this 
case  as  a  resistant  to  this  acceleration,  with  an  intensity 
equal  or  superior  to  the  weight  P.  If,  on  the  other  hand, 
the  motion  having  acquired  a  certain  velocity,  uniform  or 
accelerated,  is  suddenly  checked,  the  body  will  turn  in 
the  direction  of  the  motion.  The  inertia  of  the  body  has 
acted  here  as  a  power,  opposed  to  a  change  of  motion, 
with  an  intensity  equal  or  superior  to  the  weight  P.  In- 
ertia having  in  both  cases  produced  the  same  effect  as  the 
weight  P,  we  may  regard  it  also  as  a  force.  In  the  com- 
munication of  rapid  motion,  which  spirited  horses  impress 
upon  a  wagon,  it  is  the  inertia  of  the  wagon  which,  by  its 
resistance,  causes  the  breaking  of  the  shafts,  tug-poles,  &c. 
The  same  cause  upsets  a  wagon,  when,  impressed  with  a 
1* 


10  FOECE8   AND  THE  MEASURE  OF   THEIR   WORK. 

rapid  motion,  it  experiences  a  sudden  check  in  turning. 
It  is  this  which  throws  passengers  into  space,  when  placed 
upon  the  top  of  a  train  of  cars  suddenly  arrested  in  its 
course ; — which  breaks  tow-lines,  by  which  vessels  are 
kept  from  being  borne  too  rapidly  away  by  the  current — 
iron  anchors,  and  cables  of  ships  to  which  the  wind  and 
waves  have  imparted  a  great  velocity — balls  which  pen- 
etrate masonry,  earth  and  sand,  though  softer  than  iron 
— teeth  and  gearing,  when  suddenly  connected  with  heavy 
machines,  such  as  for  boring  cannon,  for  powder-mills, 
&c.,  &c. 

Pupils  should  be  directed  to  search  themselves  for  ex- 
amples of  these  effects  of  inertia,  as  a  motive  or  resisting 
force,  that  they  may  be  familiar  with  its  existence  and  in- 
fluence on  various  movements. 

7.  Mode  of  action  of  forces. — Forces  always  act  grad- 
tially,  and  may  be  constant  or  variable,  but  always  con- 
tinuous or  progressive  during  a  period  of  certain  duration. 
This  action  manifests  itself  sometimes  very  slowly,  and  by 
insensible  degrees,  at  others  with  rapidity,  but  never  in- 
stantaneously. 

If  the  phenomena  ever  take  place,  in  intervals  of  time, 
inappreciable  to  our  senses  and  means  of  observation,  the 
imperfection  is  entirely  due  to  that ;  for  the  more  sensible 
we  render  these  means,  the  better  can  we  appreciate  the 
duration  of  phenomena,  which  were  before  regarded  as 
instantaneous. 

All  bodies  being  more  or  less  compressible,  flexible  or 
elastic,  are  changed  in  form  by  the  reciprocal  action  of 
efforts  exerted  against  each  other,  varying  from  time  to 
time,  and  these  changes,  these  flexures,  greater  or  less, 
can  only  be  accomplished  in  a  definite  period  of  time. 
Even  in  the  rapid  phenomena  of  the  transmission  of  mo- 
tion by  a  shock,  the  efforts  developed,  and  the  velocities 
transmitted  or  lost,  are  all  gradual  and  continuous.  It  is 


FORCES   AND   THE   MEASURE   OF   THEIR  WORK.  11 

easy  to  find  examples,  in  the  shock  of  projectiles,  the 
game  of  tennis,  in  foot-ball,  &G. 

It  would  then  be  too  serious  a  departure  from  natural 
effects,  and  starting  with  an  hypothesis  too  contrary  to 
facts,  to  suppose,  as  has  been  done,  that  the  transmissions 
of  motion  in  shocks  are  made  instantaneously.  From 
such  a  supposition  incorrect  ideas  would  follow,  and  some- 
times consequences  totally  erroneous ;  hence  we  should 
bear  in  mind,  that  all  forces  acting  in  nature,  are  analo- 
gous and  comparable  to  tensions,  to  pressures  which  act 
gradually  and  continuously. 

8.  Measure  of  forces. — To  ascertain  the  measure  of 
forces,  we  admit  as  an  axiom,  that  two  forces  are  equal 
when,  substituted  for  each  other,  they  produce  the  same 
effects,  in  the  same  circumstances,  or  destroy  a  third  which 
is  directly  opposed  to  them. 

This  granted,  if  we  take  a  spring,  or  a  dynamometer, 
whose  flexures  under  the  action  of  known  weights, 
have  been  measured  and  observed  for  a  sufficient  range, 
and  submit  them  to  the  action  of  any  force,  when 
this  force  shall  have  produced  in  the  spring  a  flexure 
equal  to  that  due  to  a  certain  weight,  it  is  clear  that,  if 
in  the  two  cases,  the  spring  has  preserved  its  elasticity, 
the  force  and  the  weight  having  produced  the  same  effect, 
and  overcome  the  same  resistance  to  flexure,  these  two 
forces  will  be  equal.  The  weight  will  then  serve  as  a 
measure  of  the  force. 

What  has  been  said  in  relation  to  the  simple  matter 
of  measuring  force,  will  apply  to  any  effort  developed  by 
animate  or  inanimate  motors ; — exerting  efforts  of  trac- 
tion, such  as  that  of  horses,  locomotives,  towboats,  which 
can  be  realized  in  practice  by  simple  means,  to  be  de- 
scribed hereafter. 

We  admit,  then,  for  the  future,  that  all  forces  acting 
in  machines,  are  comparable  to  weights. 

9.  The  unit  of  measure  of  forces. — Forces  being  com- 


12  FORCES   AND   THE   MEASURE   OF   THEIR   WORK. 

parable  to  weights,  we  shall  adopt  a  unit  of  weight  for  a 
unit  of  measure  of  forces,  and  we  shall  express  them  in 
pounds,  simply  signifying,  that  in  the  same  circumstances 
they  produce  the  same  eifec.t  as  the  corresponding  num- 
ber of  pounds  acting  in  the  same  manner. 

10.  Different  names  of  forces. — We  distinguish  forces 
by  different  names,  according  to   the   circumstances  in 
which  they  act.     Thus  we  call  those  forces  motive  or  mov- 
ing, which  produce  or  maintain  motion ; — those  resisting 
forces  which  tend  to  check  or  retard  it ; — those  accelera- 
ting or    retarding  which  accelerate   or  retard  ; — those 
attracting  or  repelling  which  relate  to  attractions  or  re- 
pulsions.    Finally,  we  use  the  words  powers  and  resist- 
ances to  denote  those  forces  which  favor  motion,  or  those 
which  oppose  it. 

11.  The  constitution  of  bodies. — We  have    already 
stated,  that  all  bodies  in  nature  may  be  considered  as 
composed  of  elements,  of  infinitely  small  molecules.  These 
molecules  are  united  to  each  other  by  attracting  forces, 
and  at  the  same  time  held  at  certain  distances,  by  other 
forces  called  repelling.    These  are  the  forces  which  we 
term  molecular  forces,  and  which  maintain  the  body  or 
parts  composing  it,  in  its  form,  so  long  as  no  other  cause 
interferes  to  alter  it. 

When  repulsive  and  attractive  forces  have  great  in- 
tensity, bodies  resist  with  energy  every  cause  or  force 
which  tends  to  change  them,  and  they  are  called  solids. 
But  the  denomination  of  these  forces  is  relative  rather 
than  absolute,  and  bodies  which  we  call  liquid  or  gaseous 
are  constituted  like  those  of  which  we  have  spoken  and 
named  as  solids.  All  the  difference  consists  in  the  fact, 
that  the  molecular  attractive  forces  preponderate  in  solid 
bodies,  that  they  maintain  the  particles  in  a  closer  union, 
and  oppose  to  their  division  and  separation,  a  greater  en- 
ergy than  liquids,  whose  facility  in  separation,  movement, 
and  disjunction,  under  the  most  feeble  efforts  of  action, 


FOKCES   AND   THE   MEASURE   OF    THEIR   WORK.  13 

seems  to  indicate  a  near  equality  between  the  attracting 
and  repelling  forces.  Finally,  in  gases,  the  repelling 
forces  prevail  over  the  attracting,  and  these  bodies  of  them- 
selves tend  to  occupy  greater  volumes,  as  the  obstacles 
surrounding  them  become  weaker. 

It  follows  from  these  considerations,  that  properly 
speaking  there  is  no  determinate  limit  between  solids, 
liquids,  and  gases,  which  are  similarly  constituted,  and 
of  common  properties,  and  we  must  not  lose  sight  of  the 
fact  that  the  molecules  or  material  parts  composing  them 
may  be  united  or  separated  under  the  action  of  exterior 
forces.  These  ideas  conforming  to  the  precise  nature  of 
the  bodies,  exclude  all  notions  of  hardness  or  INFLEXI- 
BILITY, or  of  soft  bodies  deprived  of  every  faculty  of  re- 
turning, either  partially  or  completely,  to  their  primitive 
forms,  or  of  all  elasticity. 

12.  The  principle  of  action  equal  and  opposite  to  re- 
action.— From   what  precedes,  we  can  easily  conceive, 
that  when  two  bodies  press,  impinge,  or  draw  upon  each 
other,  there  is  developed  at  the  points  of  contact,  on  one 
part  efforts  of  compression  or  extension,  and  on  the  other 
efforts  of  repulsion  and  resistance,  opposite  and  equal. 
The  compressed  particles,  the  molecular  springs  bent  or 
extended,  react  with  a  force  precisely  equal  and  opposite 
to  that  compressing  or  bending  them.     It  is  the  same 
with  attractive  or  repulsive  action  exerted  any  distance, 
so  that  the  molecules  composing  them  attract  and  repel 
each  other  with  precisely  equal   and  opposite  energies. 
These  reciprocal  effects  constitute  one  of  the  fundamental 
principles  or  axioms  of  Mechanics,  and  we  declare,  in  the 
words  of  their  author  Newton,  "  that  action  and  reaction 
are  alwags  equal  and  opposite"  that  is  to  say,  " the  ac- 
tion of  two  bodies  upon  each  other,  are  always  equal  and 
in  opposite  directions."     (Third  Law). 

13.  Examples  of  this  law. — (The   case  of  two  men 


4  FOKCES  AND   THE   MEASURE  OF  THEIK  WOKE. 

pulling  at  the  ends  of  the  same  rope ; — penetration  of  pro- 
jectiles, and  the  reaction  from  the  part  penetrated.) 

14.  Point  of  application  of  forces. — The  action  of  a 
force  upon  a  body,  can   only  be  transmitted  gradually 
from  the  point  of  its  immediate  application  to  the  inte 
rior,  by  a  succession  of  flexures  or  molecular  springs  ;  and 
as  we  have  said  in  (§  7),  a  certain  time  is  needed  for  the 
operation  of  this  transmission.     When  the  force  becomes 
constant,  a  state  of  equilibrium  is  produced  between  it 
and  the  springs  it  bends,  and  from  that  instant,  if  the 
equilibrium  continues,  we  may  regard  the  bodies  as  rigid 
and  inextensible.    Now  in  machines  we  always  make  use 
of  bodies  of  such  small  flexibility,  and  so  proportioned, 
that  the  efforts  to  which  they  are  submitted  bend  them 
so  slightly  that  we  may  neglect  the  effects  of  flexure, 
which  seldom  show  themselves  in  a  sensible  manner,  ex- 
cept at  the  commencement  of  action  or  motion  ;  and  in 
all  such  cases  we  may  regard  the  bodies  as  rigid,  and  the 
efforts  as  transmitted  in  their  own  direction,  and  through 
any  point  of  this  direction,  invariably  connected  with  the 
true  point  of  application. 

But  when  there  are  shocks,  and  variable  efforts  caus- 
ing frequently  recurring  compressions,  we  shall  find  losses 
of  effect  resulting  from  them,  which  must  be  taken  into 
account.  This  reservation  evidently  applies  to  soft  bodies, 
whose  forms  are  changed  under  the  action  of  external 
forces. 

15.  Effect  and  work  of  forces. — From  what  precedes 
it  follows,  that  from  the  instant  the  force  has  commenced 
acting,  it  produces  flexures  and  compressions  in  the  direc- 
tion of  its  action ;  and  that  its  immediate  point  of  appli- 
cation yields,  moves,  and  is  displaced  in  the  direction  of 
this  action,  until,  the  resistance  of  the  molecular  springs 
being  equal  to  the  force  tending  to  bend  them,  this  rela- 
tive displacement  ceases. 


FORCES    AND   THE   MEASURE   OF   THEIR   WORK.  15 

Then>  if  the  body  is  held  by  obstacles  and  superior 
resistances,  the  action  of  the  force  is  cancelled,  since  it 
has  produced  no  motion.  Such  is  the  case  with  a  sup- 
port, a  column,  a  man  sustaining  a  burden,  horses  which 
cannot  start  a  mired  wagon,  and  with  overpressed  rollers 
which  cannot  overcome  the  resistance  of  iron. 

For  these  forces  to  produce  a  mechanical  or  industrial 
effect,  or  useful  work,  they  must  have  passed  through  a 
certain  path  in  their  own  direction,  at  their  point  of  ap- 
plication. Tims,  a  fundamental'  condition  of  the  mechan- 
ical or  industrial  work  of  forces,  is,  that  there  must  be,  at 
the  same  time,  an  effort  exerted,  and  a  path  described  in 
virtue  of  this  effort. 

16.  Measure  of  work  of  a  constant  force,  where  the 
path  described  ~by  its  point  of  application  is  in  its  own 
direction. — It  is  evident  that  the  effect,  the  work  produced 
by  force,  is  proportioned :  1st.  To  the  intensity  of  the 
effort :  2d.  To  the  space  described,  and  consequently  to 
the  product  of  these  two  factors.  Thus  in  raising  burdens, 
or  minerals ;  in  the  draught  of  wagons  and  ploughs  ;  in 
towing  boats,  and  drawing  water,  it  is  evident  that  for 
the  same  weight  or  effort,  the  effect  is  doubled  when  the 
space  is  doubled ;  and  that  for  the  same  space  the  effect 
is  double  or  triple,  if  the  resistance  is  double  or  threefold 

Comparing  efforts  with  weights,  whose  action  will 
produce  the  same  effect,  and  the  spaces  described  being 
expressed  in  feet,  we  see  that  the  work  of  a  constant  force 
may  be  represented  by  the  product  of  its  intensity,  (ex- 
pressed in  units  of  weight,  or  in  pounds,)  into  the  space 
described  in  its  own  direction,  expressed  in  units  of 
length,  or  in  feet.  If  we  take  for  the  unit  of  work,  the 
pound  raised  one  foot,  then  the  work  of  a  force  F,  whose 
point  of  application  has  described  the  path  S,  will  be  ex- 
pressed by  FS  pounds  raised  one  foot,  which  we  may 
designate  by  Ib.  ft.,  written  a  little  above  the  right  of  the 
product  FS.  Thus  FS lb8- ft- 


16  FORCES   AND   THE   MEASURE   OF   THEIR   WORK. 

IT.  Representation  of  this  work  ly  the  surface  of  a 
rectangle. — If  we  take  the  space  S  for  the  base  of  a  rec- 
tangle, whose  height  at  a  certain  scale  shall  be  the  effect 
F,  it  is  evident  that  the  product  FS  will  be  the  measure 
of  the  surface  of  the  rectangle ;  or  that  reciprocally  this 
surface  may  be  taken  to  represent  the  work  FS. 

18.  Measure  of  the  work  of  a  variable  force* — When 
a  force  is  variable,  we  may  apply  the  same  method  of 
measurement  to  each  of  the  small  elementary  spaces,  s,  in 
which  we  may  consider  the  force  as  constant.  The  work 
corresponding  to  each  of  these  elementary  spaces  is  repre- 
sented then  by  the  product  FS. 

If  we  place  upon  the  straight  line  AB  taken  for  the 
axis  of  abscissae,  the  spaces  described,  and  at  each  point 

of  division  raise  a  perpendicu-         ^ — n ~^_^ 

lar    representing  at  a   certain   ^"l""" 

scale,  the  effort  exerted,  we  shall       i  I 

then  have  a  curved  surface  limi-  -^j  j 

ted  by  the  line  of  abscissa,  the 

extreme  ordinates,  and  the  curve    -J b 

passing  through  the  extremities    -^  s  •" 

of  all  the  ordinates.  If  we  consider  the  small  elementary 
trapezium  as  corresponding  with  any  effort  F,  and  with 
an  element  of  the  space,  s,  it  is  clear  that. the  surface  of 
this  small  trapezium  will  be  Fs,  and  that  it  will  represent 
the  elementary  work  corresponding  to  the  small  space  s. 

The  whole  work  for  a  space  s  being  composed  of  the 
sum  of  all  the  elementary  quantities  of  work  Fs,  it  is  evi- 
dently represented  by  the  whole  surface  limited  by  the 
curve.  We  have  only  then  to  find  this  surface,  or  the 
sum  of  all  the  elementary  products  Fs.  Calculation  in 
certain  cases  affords  direct  methods  of  obtaining  it — t>ut 
in  many  others,  and  in  practice,  it  is  best  to  employ  the 
method  of  quadrature,  particularly  that  of  Simpson,  which 
has  been  already  explained. 

Moreover,  it  is  absolutely  indispensable  to  recur  to 


FORCES   AND   THE   MEASTTKE    OF   THEIR   WORK. 


17 


these  methods  when  we  wish  to  estimate  work  transmitted 
by  animal  motors,  and  by  many  machines,  in  which  the 
efforts  transmitted  are  constantly  varying,  according  to 
laws  impossible  to  be  found. 

19.  Application  to  work  developed  by  horses  hauling 
a  mail-boat  on  the  canal  de  rOurcq. — By  means  of  appa- 
ratus, to  be  described  hereafter,  we  have  obtained  an  ex- 
perimental curve,  or  graphic  relation  between  the  space 
described  and  the  efforts  exerted.  It  would  be  impossi- 
ble by  any  direct  method  of  calculation,  to  obtain  the 
relation  existing  between  the  efforts  and  spaces  described, 
for  deducing  the  work ;  but  the  rule  of  quadrature  fur- 
nishes us  the  means.  Operating,  for  example,  over  a 
space  of  157.48 ft>  in  length,  which  we  divide  into  twelve 
equal  parts,  we  find  for  the  ordinates  F15  Fa,  .  .  .  .  F12,  F,8 
the  following  values,  according  to  the  scale  of  flexures  of 
the  spring : 


FFFFFFFFFF   F    FF 

x      -1-     A      A      ^      ±      -LJ--L-t-L-L-L 


F,  =191.9  F2  =275.08 

F13=283.4  F4  =266.92 

475.3  F6  =200.08 

F8  =241.55 

F10=158.38 

F12=187.51 


F3  =258.10    S, 

F6  =216.84    I2~    12 

F7  =208.46 

F9  =208.46 

Fn=158.38 

1050.24x2=2100.48 


1329.52x4=5318.08 


18  FOECES   AND   THE   MEASURE   OF   THEIK   WOEK. 

The  whole  work  for  this  space  is  then 

I  x  13.12(475.3+5318.08+2100.48')=34522.4lbs- ft- 

This  experiment  was  made  with  a  boat  weighing,  in- 
clusive of  load,  15766 lbs-  moving  at  the  rate  of  15.45 ft- 
per  second,  or  10^  miles  per  hour. 

20.  Indretfs  Steam  Engine. — The  diameter  of  the 
piston  being  1.18112  "••  its  surface  =1.181 12  x  0.7854= 
1.0958  sq-ft-=157.766s<1-ins- 

The  whole   stroke  is   3.0184 ft-     Dividing  it  into  16 

1  S 

equal  parts  we  have  -  —=0.062883  ft- 
3  lo 

The  abstract  of  the  curve  of  pressure  furnished  by  the 
index,  gives  the  following  result  for  the  pressiire  upon 
each  0.00107643  sq-ft-  of  surface  of  piston. 


FFFFFFFF 

•»•  i   •*•  a  J-  a    J-4    J-s-^e    x  7   x 


lbs. 


lbs. 


148.35165 


F,  =.425657  Fa  =2.43043  F3  =3.49567        Fa-}-Fn  =  0.88219 

F17=. 456534  F4  =4.25657  F6  =4.33375  4(F2..+F16)=99.35212 

.882191 F6  =4.37786  F7  =4.37786  2(F3  ..+F15)=48.11734 

F8  =4.37786  F9  =4.37786 

F10=4.25657  Fn=3.79702 

F12=2.50982  F13=2.27825 

F14=1.59676  F15=1.39826 

F16=1.03216        24.05867  x  2=48.11734 

24.83803x4=99.35212 


FORCES   AND   THE   MEASURE   OF   THEIR   WORK.  19 

And  for  the  work  developed  by  the  steam  in  one  stroke, 
.062883 ft-  x  1018  x  148.3516 lbs~ 9498.4 lbs- ft- 

21.  Mean  effort  of  a  variable  force. — It  is  often  useful, 
and  even  necessary  to  ascertain  the  mean  effect  of  a  va- 
riable force  ;  that  is,  the  constant  effort  which  would  pro- 
duce the  same  work,  causing  it  to  pass  through  the  same 
space  at  its  point  of  application.  From  this  definition 
and  the  preceding  remarks,  if  we  call  W  the  work  devel- 
oped by  the  variable  force,  and  S  the  whole  space 
described  by  the  point  of  application,  we  shall  have 

W 

W=FS,  whence  F=-5-.     Thus  we  shall  obtain  the  mean 

b 

effort  of  a  variable  force  in  dividing  the  total  work  by 
the  space  described. 

It  follows  from  this 
that  the  work  of  the  va- 
riable force  being  repre- 
sented (Fig.  8)  by  the 
area  AabcdefM,  the  work 
of  the  corresponding  con- 
stant mean  force  will  be 
represented  by  the  sur- 
face  of  the  rectangle  FIG.  s. 

AA'M'M,  having  the  same  area  as  the  curve.  We  may 
here  remark  that  the  points  5,  c,  and  0,  where  the  curve 
of  variable  effort  cuts  the  line  A'M'  of  the  mean  constant 
effort,  correspond  to  the  positions  where  these  two  efforts, 
as  well  as  the  elementary  work  developed  by  them  are 
equal.  Moreover,  the  areas  A!ab  and  cde  above  the 
straight  line  A'M7,  represent  the  excess  of  the  work  of 
the  variable  effort  above  that  of  the  constant  effort  while 
the  body  passes  through  the  space  AB  and  CE;  in  the 
same  way  the  areas  contained  between  the  line  A'M'  and 
the  curve  below  it,  represent  the  excess  of  work  of  the 
constant  effort,  above  that  of  the  variable  force.  The 


20  FOKCES    AND   THE   MEASURE   OF   THEIR    WORK. 

sum  of  these  first  excesses  should  evidently  equal  the  sum 
of  the  second. 

22.  Observations  upon  the  mode  of  calculation  adopted 
in  English  practice. — Some  authors,  and  particularly  the 
practical  English  engineers,  in  calculating  the  effect  of 
steam  engines,  often  take,  for  the  mean  effort,  the  arith- 
metical mean  of  the  extreme  pressures  or  forces,  and  mul- 
tiply it  by  the   space  described.     If,  for  example,  the 
work  developed  by  the  steam  during  its  expansion  were 
required,  the  curve  which  gives  the  effort  corresponding 
to  each  stroke  of  the  piston,  as 

we  shall  see,  and  as  the  figure 
indicates,  would  be  convex  to- 
ward the  line  of  abscissae  ;  and 
taking  the  arithmetical  mean 
between  the  extreme  ordinatesv 
and  multiplying  it  by  ac,  we 
have  the  area  of  the  trapezium 
abdc,  much  greater  than  that  Fl0t  9. 

of  the  curve. 

23.  The  case  where  the  arithmetical  mean  of  a  certain 
number  of  variable  values  may  be  taken  for  the  mean 
effort. — When  the  values  of  the  effort  oscillate  periodi- 
cally around  some  determinate  effect,  or  between  certain 
limits  very  numerous,  taken  independently  of  the  irregu- 
larity of   their  periods  of  oscillations,  the   arithmetical 
mean  of  a  great  number  of  these  values  may  be  taken  for 
the  mean  effort,  with  sufficient   accuracy  for   common 
practice.     This  is  particularly  the  case  in  experiments 
upon  the  action  of  animal  motors,  and  in  the  efforts  trans- 
mitted by  various  manufacturing  machines,  as  will  be  seen 
in  some  following  examples. 

24.  Applications. — We  have  seen  in  an  experiment 
cited  in  (§19),  that  the  work  developed  by  three  horses 
harnessed  to  a  mail  boat,  was  34522.4 lbs- ft-  for  a  space  of 


FORCES   AND   THE   MEASURE   OF   THEIR   WORK.  21 

157.48 ft-    The  mean  effort  to  produce  the  same  work 
would  be 

MS^-". 

157.48"- 


or  for  each  horse       - =73.06 lbs- 
3 

So,  also,  in  the  example  relating  to  work  developed  upon 
the  piston  of  the  engine  of  the  millwright's  shop  at  In- 
dret,  we  have  found,  for  a  stroke  of  3.0184 ft-  the  total 
work  to  be  9498 lbs>  The  corresponding  mean  effort 
would  be 

9498.4 lbs- ft- 


lbs- 


3.0184 ft- 


=3146 


The  surface  of  the  piston  being  1  57.76  sq>  In"  this  mean 
effort  corresponds  to  a  pressure  of 

3146  lbs* 

sq.  in.  =19-9  lbs*  Per  sauare  inch. 


25.  The  idea  of  work  is  independent  of  time.  —  We  see 
from  what  precedes,  that  in  the  measure  of  work,  we 
have  only  regarded  the  effort  exerted,  and  the  space  de- 
scribed in  the  direction  peculiar  to  this  effort.  It  is 
therefore  independent  of  time. 

Thus  in  raising  goods,  the  effect  is  not  measured  by 
the  duration  of  labor,  but  by  the  product  of  the  load  into 
the  height  of  its  elevation. 

Still,  when  the  work  is  of  long  duration,  and  is  peri- 
odically repeated  in  the  same  manner,  the  measure  of  one 
determined  period  is  evidently  sufficient  to  ascertain  any 
other.  It  is  thus,  in  the  periodic  action  of  steam  engine 
work,  of  hydraulic  wheels,  of  animal  motors,  that  we  re- 
fer the  work  to  a  unit  of  time,  which  we  usually  make 
equal  to  a  day,  an  hour,  a  minute  or  a  second.  (This  last 
unit  is  most  frequently  used.) 

For  animal  motors,  whose  work  is  limited  by  fatigue, 


22  FOECES   AND   THE   MEASURE   OF   THEIR   WORK. 

and  the  need  of  rest,  in  our  estimate  of  its  value  per  sec- 
ond, we  must  regard  the  whole  duration  of  the  work,  since 
it  has  great  influence  upon  its  value  in  each  unit  of  time. 
Thus  a  strong  wagon-horse  may  travel  from  8  to  10  hours 
per  day,  developing  in  each  step  at  a  velocity  of  3.28 f *• 
per  second,  from  434  to  480lbs-ft-  of  work;  while  horses 
employed  in  hauling  mail  boats,  which,  in  the  case  we 
have  cited,  (19,)  developed  a  mean  effort  of  73.06  lb%  run- 
ning at  the  rate  of  15.45 ft-  per  second,  and  giving  the 
value  of  work  as  73.06lbs-xl5.45=1129lbs-  f%  can  only 
travel  two  hours  at  most,  with  four  relays  per  day,  resting 
one  day  in  four,  and  yet  rapidly  wearing  out  in  this  hard 
service. 

26.  Various  denominations  of  mechanical  work. — The 
mechanical  effect  of  forces,  which  we  shall  measure  by 
the  product  of  the  effort  into  the  space  traversed  in  its 
peculiar  direction,  has  received  different  names,  which  it 
is  well  to  know. 

Smeaton,  the  English  engineer,  to  whom  we  are  in- 
debted for  useful  experiments  on  water-wheels,  and  wind- 
mills, called  it  mechanical  power  ;  Carnot,  the  moment  of 
activity  ;  Monge  and  Hachette,  dynamic  effect ;  Coulomb 
and  M.  Navier,  quantity  of  action  •  MM.  Coriolis  and 
Poncelet,  quantity  of  work.  We  shall  adopt  the  last  ex- 
pression, as  most  appropriate  to  the  industrial  view  which 
we  shall  take  of  mechanics. 

27.  Unit  of  mechanical  work. — As  to  the  value  of  the 
unit  of  work,  we  have  said  that  we  shall  adopt  the  pound 
raised  one  foot.     Some  French  authors  have  proposed  for 
a  unit  of  work,  1000 kil°s— 2205 lbs-  raised  lmetre=3.28ft- 
in  height,  giving"  it  the  name  of  dyname  or  dynamode. 

Another  unit  which  has  come  into  use,  notwithstanding 
its  faulty  denomination,  is  what  is  termed  horse-power. 
This  expression,  introduced  by  Watt,  at  a  time  when  the 


FORCES   AND   THE   MEASURE   OF   THEIR   WORK.  23 

steain-engine  had  been  successfully  substituted  for  horse- 
power, expresses  a  work  equivalent  to  33000  lbs-  avoirdu- 
pois raised  lft-  per  minute,  equal  to  550  lbs-  per  second. 
The  value  generally  adopted  in  France  is  542.7  lbs> 

Though  this  estimate  of  the  horse-power  is  at  present 
used  as  a  conventional  unit,  it  has  no  legal  value,  though 
it  is  very  desirable  that  a  legislative  act  may  give  it  this 
character,  for  it  is  the  money  of  industrial  work. 

"We  need  hardly  add,  that  this  expression  has  no  direct 
relation  to  the  work  actually  developed  by  horses  tackled 
to  gins,  which  seldom  exceeds  a  mean  of  from  289  to 
325  ibs.  ft.  per  second. 

Example.  —  In  the  experiment  relative  to  the  steam- 
engine  at  Indret,  where  we  found  the  work  developed  by 
the  steam,  in  a  stroke  of  the  piston  equal  to  9496  11)S-  f% 
there  were  28  double  strokes  per  minute,  so  that  the  work 
per  second  would  be 

v  Kfi 

—  6=8863lbs-ft-, 


60 
and  the  force  in  horse-power  would  be 

8863lbs.ft. 


550 


=16  horse-power. 


28.  Observations  upon  the  conditions  of  mechanical 
work. — "We  have  said,  that  the  work  of  a  force  will  be 
measured  by  the  product  of  its  intensity,  into  the  space 
traversed  in  its  proper  direction,  but  it  should  be  under- 
stood that  this  space  is  described  by  the  effect  of  the  force 
itself.  Thus  a  man  in  a  boat  or  rail-car,  who  exerts  a 
force,  in  the  direction  of  motion,  upon  an  object  which 
receives  from  it  no  relative  motion,  will  not  produce  any 
useful  work,  although  the  body  may  move  in  the  direction 
of  the  effort,  by  the  general  effect  of  the  motion  trans- 
mitted. 


24  FORCES   AND   THE   MEASURE   OF   THEIR   WORK. 

It  is  so  in  case  the  effort  is  perpendicular  to  the  path 
described ;  there  will  then  be  a  pressure,  an  effort,  but 
no  work  produced  by  the  effort.  From  this  cause  we 
have  disturbances,  and  friction,  causing,  as  we  shall  see, 
losses  of  work,  but  not  of  immediate  useful  effect.  "We 
would  also  remark  here,  that  the  definition  of  the  work 
of  any  force  applies  as  well  when  the  path  described  by 
the  point  of  application  of  the  force,  is  in  an  opposite  di- 
rection to  that  of  the  force,  as  when  it  is  in  the  same 
direction.  We  speak  of  the  latter  case, — the  point  of  ap- 
plication following  the  direction  of  the  force,  as  develop- 
ing a  motive  work ;  and  of  the  former, — the  point  of 
application  moving  in  a  direction  opposite  to  that  of  the 
force,  as  developing  a  resisting  work. 

29.  Horizontal  transportation  of  loads. — This  kind  of 
work  is  not  measured  by  the  method  we  have  adopted, 
since  it  produces  certain  effects,  and  expenditures  of  work, 
depending  less  upon  the  weight  transported,  than  upon 
the  mode  of  transportation.  Thus,  the  transportation  of  a 
weight  of  2205 lbs-  by  means  of  a  sledge  slipping  along  the 
ground,  where  the  friction  equals  T\  of  the  pressure, 
would  require  per  yard  passed  over,  a  work  of  661.5lbs-  x 
1 yd- ;  while  that  by  a  wagon  of  common  dimensions,  where 
the  draught  is  ¥V  the  load,  would  require  a  work  of 
76.83 lbs>  xlyd- ;  and  that  by  a  railway  car  at  a  small  ve- 
locity— the  resistance  being  but  ^¥  of  the  load — would 

2205 
require  a  draught  of  — —  :=7.35lb%  and  the  wrork  per  yard 

would  be  7.35  lbs-yds-. 

We  see,  then,  that  the  work  relative  to  the  horizontal 
transportation  of  loads,  cannot  be  measured,  as  we  have 
hitherto  done,  by  the  product  of  the  weight  into  the  path 
described ;  but  rather  by  a  comparison  of  results  made 
for  the  particular  service,  and  kind  of  transportation. 


FORCES   AND    THE   MEASURE    OF    THEIR   WORK. 


25 


30.  Case  where  the  force  does  not  act  in  the  same  di- 
rection as  the  path  described. 

If  the  path  described  is  Aa,  _F 

while  the  direction  of  the  force 
is  AF,  it  is  clear  that  the  path 
described  in  the  direction  of  the 
force  will  be  determined  by  the  _ 
perpendicular  a5,  let  fall  from  a 

upon  AF,  and  equal  to  AJ).    The  work  developed  by  the 
force  F  will  then  be,  according  to  the  definition,  F  x  A5. 

This  may  be  otherwise  readily  understood  from  a  con- 
sideration of  the  adjoining  figure. 

Let  AB  be  the  direction  of  a 
force  P,  acting,  at  a  certain  in- 
stant, upon  a  body  describing  the 
curve  LM,  when  the  body  is  sup- 
posed to  have  arrived  at  A.  If 
we  conceive  the  line  AB  to  be  an 
inextensible  and  perfectly  flexi- 
ble thread,  and  the  action  of  the 
force  P  to  be  replaced  by  a 
weight  Q,  acting  at  the  end  of 
this  thread,  which  rolls  over  a 

pulley  o,  perfectly  movable  around  its  axis ;  it  is  clear, 
that  in  the  elementary  displacement  of  the  body  from  A 
to  a,  the  work  of  the  force  P,  will  be  measured  by  the 
product  of  the  weight  Q  into  the  quantity  W  which  it 
will  have  fallen.  Now  this  quantity  W  is  equal  to  the 
difference  of  length  of  the  lines  AB  and  aB,  where  the 
point  of .  intersection  B,  may  be  regarded  as  the  point  of 
instantaneous  contact  of  the  directions  AB  and  aB  with 
the  periphery  of  the  pulley.  But  on  rolling  the  thread 
aB  upon  the  periphery,  its  extremity  a  will  describe  an 
elementary  arc  of  the  involute  aa7  perpendicular  to  AB, 
and  the  length  Ka'  will  measure  precisely  the  difference 
sought.  The  arc  aa'  being  merged  at  the  smallest  limit 


26  FORCES   AND   THE   MEASURE   OF   THEIR   WORK. 

into  a  perpendicular  let  fall  from  the  point  a  upon  AB, 
we  see  clearly  that  Aa'  is  what  is  geometrically  termed, 
the  projection  of  the  path  Aa  really  described  upon  the 
direction  of  the  force :  and  thus  it  is  evident  from  this 
figure,  that  the  elementary  work  of  the  force  P  is  meas- 
ured by  the  product  P  x  Aa',  of  the  intensity  into  the 
projection  of  the  infinitely  small  path  Aa,  upon  its  own 
direction. 

When  the  force  is  not  in  the  direction  of  the  path  de- 
scribed, the  work  due  to  the  elementary  displacement  Aa 
of  its  point  of  application,  is  the  product  of  the  intensity 
of  the  force  by  the  projection  of  the  displacement  Aa 
upon  its  own  direction.  This  product  is  what  is  termed, 
in  rational  mechanics,  the  virtual  moment,  though  we 
shall  give  it  the  name  of  elementary  work.  This  identity 
will  lead  us  to  many  analogies  with  the  results  of  rational 
mechanics,  and  this  natural  expression  of  work  will  help 
us  toward  an  easier  appreciation  of  its  demonstrations. 

31.  The  work  of  weight  upon  a  body  describing  any 
curve. 

If  we  consider  the  body  as  arrived 
at  A,  and  then  describing  the  small 
elementary  path  Aa,  the  correspond- 
ing elementary  work,  developed  by 
gravity,  whose  direction  is  always 
vertical,  will  be  the  product  of  the 
weight  of  the  body,  P,  by  the  height  ] 
Ab,  which  it  has  described  in  the  di- 
rection of  this  force.  Weight  being  constant,  for  the 
same  place,  and  for  heights  varying  but  little  at  the  earth's 
surface,  the  total  work  developed  after  the  body  has  de- 
scended from  L  to  M  will  be  the  product  of  P  by  the  sum 
of  the  projections  analogous  to  Ab,  or  by  the  total  height 
of  the  descent  II,  and  will  consequently  be  equal  to  PH. 
Whatever,  then,  may  be  the  curve  of  descent,  it  is  the 


FORCES    AND    THE   MEASURE    OF    THEIR   WORK.  27 

same,  and  depends  only  upon  the  difference  of  level  of 
the  extremities  of  this  curve. 

32.  The  crank  and  its  connecting-rod. — When  the  arm 
of  a  crank  is  sufficiently  long  for  us 

to  disregard  its  obliquities,  it  is  clear, 
if  the  effort  exerted  in  its  direction 
is  constant,  that  the  total  work  de- 
veloped during  a  semi-revolution 
will  be  the  product  of  the  constant 
effort  F,  into  the  sum  of  the  pro- 
jections of  its  elementary  arcs  Aa 
upon  its  direction,  a  sum  evidently  equal  to  the  diameter 
2R.  Consequently,  the  work  developed  in  a  semi-revolu- 
tion is  F  x  2E. 

33.  Observation  respecting  the  direction  of  the  effort, 
in  its  relation  to  that  of  the  path  described. — If  the  path 
described  is  in  a  direction  contrary  to  that  of  the  effort,  it 
is  evident  that  the  body  is  impelled  by  another  force,  in 
relation  to  which  the  effort  F  is  a  resistance  overcoriie  ; 
we  say,  then,  that  the  work  of  a  force  F  is  resistant,  sub- 
tractive,  or  negative, — that  is,  it  must  be  deducted  from 
the  motive  work,  a  part  of  which  it  has  consumed  and 
absorbed. 

Thus,  when  a  body  descends  by  the  action  of  gravity, 
its  path  being  in  the  direction  of  the  force,  it  acts  as  a 
power,  and  its  work  is  positive ;  but  when  the  body  as- 
cends, the  path  is  in  an  opposite  direction  to  that  of  the 
force,  which  acts  as  a  resistance,  and  the  work  is  negative. 
If  the  body  descends  and  ascends  alternately  the  same 
height,  the  motive  work  developed  during  the  descent  is 
equal  to  the  resistant  work  consumed  during  the  ascent, 
and  the  total  work  is  nought.  There  is  then  an  alternate 
production  and  consumption  of  work,  in  all  cases  where 
the  bodies  periodically  ascend  and  descend,  as  in  cranks, 
pistons,  pendulums,  &c. 


28  FOECES   AND   THE   MEASURE   OF  THEIK   WORK. 

34.  Springs. — A  consumption  of  work  is  produced  also 
in  the  flexure  of  springs,  and  a  restitution  is  made  in  their 
return  to  their  primitive  form.   It  is  complete,  if  the  spring, 
in  unbending,  recovers  exactly  the  form  it  had  before :  it 
is  incomplete,  and  a  consumption  of  work  occurs,  when-' 
ever  it  but  partially  returns  to  its  primitive  form. 

35.  Expansion  and  Contraction. — It  is  the  same  also 
with  a  body  dilated  by  the  action  of  heat,  and  the  enor- 
mous efforts  developed  in  this  case  are  similar  to  those 
produced  by  other  causes.     Indeed,  we  know  by  expe- 
rience, that  bodies  expand  or  contract  between   certain 
limits,  by  quantities  proportional  to  the  efforts  to  which 
they  are  submitted.     Thus,  for  example,  a  bar  of  iron  ex- 
pands or  contracts  by  a  quantity  I,  which  expressed  in 
feet,  is  given  in  the  formula 

Plbs. 
, 


28457800 

calling  P  the  load  per  square  inch  of  section,  and  I  the 
expansion  per  running  foot. 

Reciprocally,  when  a  bar  contracts,  it  exerts  an  effort 
equal  to  the  force  required  to  produce  the  same  contrac- 
tion, and  this  effort  will  depend  upon  the  expansion  per 
running  foot. 

If,  for  example,  a  bar  of  iron  1.1811ins-  square  expands 
a  quantity  1= 0.0005 ft-  per  foot,  the  effort  capable  of  pro- 
ducing this  elongation  will  be 

P=284r5r800x  0.0005  ft-=14228.9lbs-  per  square  inch, 


or  in  all  1.18112  x  14228.9=19844 lbs- 

Observing  now  that  between  32°  and  212°  a  bar  of 
iron   expands  0.0012205 ft-  per  foot,  it  follows  that  the 


FORCES   AND   THE   MEASURE   OF   THEIR   WORK.  29 

quantity  required  to  expand  it  0.0005  ft-  per  foot  will  be 
found  by  the  proportion 

0.0012205"-  :  180°  :  :  0.0005"-  :  a?= 


Thus  by  an  increase  solely  of  the  temperature  of  the 
bar  by  about  74°,  we  may  bring  to  bear  against  obstacles 
opposing  its  expansion,  an  effort  of  19844-  pounds. 

Keciprocally,  if  this  bar,  after  heating  and  expansion, 
is  cooled,  it  exerts  efforts  of  traction  depending  upon  the 
degree  of  cooling.  In  case  of  the  reduction  of  the  tem- 
perature of  a  bar  1.18ins-  square,  by  about  74°,  there 
would  be  an  effort  of  contraction  exerted  equal  to  19844 
pounds. 

This  important  property  of  bodies  exerting  considera- 
ble efforts  of  expansion,  and  of  contraction  or  shrinkage, 
is  often  advantageously  used  in  the  Arts.  The  tires  of 
wheels,  naves,  and  the  shafts  of  water-wheels  ;  the  gird- 
ling of  domes,  particularly  that  of  the  cupola  of  St.  Peter's 
at  Rome,  are  examples  of  its  use. 

It  is  said  that  the  righting  of  the  walls  of  the  ancient 
Conservatory  Library  was  effected  by  similar  means  with 
great  success.  The  bars  used  were  2.3622  x  0.86615in% 
having  a  section  of  2.04:6  square  inches.  They  were 
heated  by  means  of  suspended  gridirons,  and  as  they  ex- 
panded, were  held  in  place  by  strong  screw  nuts  with 
cast-iron  washers  ;  they  were  then  left  to  cool. 

If,  for  example,  their  temperature  was  reduced  73.°74r, 
the  contraction  would  be  0.0005  ft-  per  foot,  and  the  cor- 
responding effort  would  be  14:228  pounds  per  square  inch  ; 
the  effort  exerted  by  each  bar  would  therefore  be 

,     U228.90  x  2.046=29112lbs-. 

As  to  the  work  developed  by  this  force,  it  is  easily 
calculated.  In  fact,  from  32°  to  212°,  and  even  beyond 


30  FOECES   AND   THE  MEASUEE   OF   THEIE   WOEK. 

this,  experiments  prove  that  the  elongations  are  propor- 
tional to  the  temperatures,  so  that  If  representing  the  ex- 
pansion up  to  212°,  and  1  that  relative  to  T°,  we  have 

T  :  180°  :  :  I  :  T; 

whence 

T     rT_0.0012205T 

~180~     ""180™ 

If  we  call  L  the  length  of  the  bar  at  the  temperature 
of  32°,  this  length  will  increase  per  lineal  foot,  in  passing 
to  the  temperature  T,  by  the  quantity  I=KT  and  will 
become 


Also,  in  passing  from  the  temperature  32°  to  the  tem- 
perature T',  the  length  of  the  bar  will  become 


The  expansion  of  the  bar,  in  passing  from  the  temper- 
ature T  to  that  of  T  will  then  be  L'—L=L1K(T'--T), 
and  the  expansion  per  lineal  foot  will  be 


whence  we  see  that  the  expansion  per  lineal  foot  depends 
upon  the  difference  of  the  temperatures,  and  not  upon 
their  particular  elevations.  Consequently,  it  is  the  same 
with  the  force,  P=I  x  28457800  lbB-=284:57800K(T/--  T)  ; 
which  increases  proportionally  with  the  differences  of 
temperatures,  and  is  the  same  for  equal  differences. 

This  granted,  if  we  place  upon  a  line  of  abscis&a,  start- 
ing from  32°,  the  expansions  Lx—  L=L1K(T'-T),  which 
at  first  are  nothing,  for  T'=T15  and  at  the  resulting  points 
of  division  erect  perpendiculars  or  ordinates  equal  to  the 


FORCES   AND   THE   MEASURE    OF   THEIR    WORK.  31 

efforts  of  expansion  or  contraction,  which  have  for  their 
values  those  of  the  force 

P=284:57800.     I=28457800K(T/-T), 

it  is  clear  that  the  ordinates  be- 
ing proportional  to  the  abscissa, 
the  points  thus  determined  will 
be  in  a  straight  line,  and  thus 
will  form  a  triangle,  whose  sur- 
face expresses  the  work  devel-  FIG.  14. 
oped  by  the  efforts  of  expansion  or  contraction,  corre- 
sponding to  the  different  temperatures  T'— T. 

The  surface  of  the  triangle  is  moreover 

i 

ft. 


So  that  the  work  developed  by  the  efforts  of  expansion  or 
contraction  has  definitely  for  its  value 

14228900K2L1(T/-T)2  lbs- ft-, 

A     iLilLi      f     TT  •*       i       0.0012205  4,  . 
and  substituting  for  K  its  value  -  -  this  expression 

lot) 

of  work  becomes 

0.0006541  eLXT'-T)2  lbs- ft-. 

It  shows  that  this  work  is  proportional  to  the  length  of 
the  bar  at  the  temperature  of  32°,  and  to  the  square  of  the 
difference  of  temperatures. 

We  also  see,  that  it  does  not  depend  upon  the  temper- 
atures themselves,  but  rather  upon  their  differences,  so 
that  for  the  same  variation  we  have  always  a  correspond- 
ing work. 

If  we  suppose  T=68°,  T'=141.°8,  we  have  T7— T=T3.°8 
and  consequently  the  work  of  the  bar,  per  running  foot 
and  per  square  inch  of  section,  is 

0.00065416  x  73.8a=3.5629 lbs-  f 


32  FORCES   AND   THE   MEASURE   OF   THEIR   WOKK. 

and  for  the  2.046  square  inches  for  a  length  of  32.808 ft-  it 
will  be 

3.5629 lbs- ft-  x  2.046  x  32.808=239.17 lbs- ft- 

36.  The  proper  limit  to  variations  of  temperature  to 
be  used. — We  have  confined  ourselves  in  the  preceding 
calculations  to  a  variation  of  temperature,  because,  as  we 
have  seen,  it  corresponds  to  an  expansion  or  a  shrinkage 
of  .0005  per  foot,  and  to  an  effort  of  14228.5 lbs-  of  exten- 
sion or  compression,  which,  from  observations  of  good 
constructions,  is  the  highest  limit  of  effort  which  forged 
iron  can  support  per  square  inch  of  section,  without  fear 
of  deranging  its  elasticity,  as  we  shall  hereafter  see.  It 
is  important,  therefore,  that  we  should  confine  ourselves 
to  limits  of  extension  or  contraction  between  which  elas- 
ticity is  not  altered. 


DYNAMOMETEES, 

OK,  THE  DESCRIPTION  AND  CONSTRUCTION  OF  INSTRUMENTS 
ADAPTED  TO  THE  MEASUREMENT  OF  WORK  DEVELOPED  BY 
ANIMATE  OR  INANIMATE  MOTORS. 

37.  General  and  particular  conditions  which  these  in- 
struments should  fulfil. — We  have  seen  in  the  preceding 
lessons,  that  the  work  developed  by  a  constant  force  F, 
(whose  point  of  application  has  described  the  path  S  in 
its  own  direction,)  had  for  its  measure  the  product  FS,  and 
that  if  the  effort  F  is  variable,  the  total  work  developed 
after  the  body  has  passed  through  any  path  S,  was  the 
sum  of  all  the  elementary  quantities  of  work  Fs,  succes- 
sively developed  along  the  elements  s  of  the  path  described. 
In  this  last  case,  we  have  shown,  either  by  calculation  or 
Simpson's  quadrature,  how  we  obtain  the  sum  of  products 
analogous  to  Fs,  for  the  total  given  path  S  described  in 
the  direction  of  the  effort.  Finally,  we  have  defined  the 
mean  effort  of  a  variable  force,  and  shown  that  the  total 
work  is  deduced,  by  dividing  the  former  by  the  total  space 
described. 

The  instruments  designed  to  measure  work  developed 
by  animate  or  inanimate  motors,  should,  then,  afford  us 
by  their  indications  the  product  of  the  effort  into  the 
space  described,  whatever  may  be  their  simultaneous  va- 
riations. Such  is  the  general  condition  to  be  fulfilled  in 
all  cases  not  involving  impossibilities. 

The  illustrious  Watt  was  the  first  to  satisfy  this  con- 
3 


34:  DYNAMOMETEES. 

dition,  in  the  construction  of  a  dynamometric  apparatus, 
to  which  he  gave  the  name  of  " Pressure  Indicator"  a 
description  of  which  will  be  given  hereafter. 

"We  will  now  consider  the  particular  conditions  to  be 
fulfilled  by  these  instruments  : 

1st.  The  sensibility  of  the  instrument  should  be  pro- 
portioned to  the  intensity  of  efforts  to  be  measured,  and 
should  not  be  liable  to  alterations  by  use. 

2d.  The  indications  of  flexures  should  be  obtained  by 
methods  independent  of  the  attendance,  fancies,  or  pre- 
possessions of  the  observer,  and  should  consequently  be 
furnished  by  the  instrument  itself,  by  means  of  tracings 
or  material  results,  remaining  after  the  experiments. 

3d.  "We  should  be  able  to  ascertain  the  effort  exerted, 
at  each  point  of  the  path,  described  by  the  point  of  appli- 
cation of  the  effort,  or,  in  certain  cases,  at  each  instant  in 
the  period  of  observations. 

4th.  If  the  experiment  from  its  nature  must  be  con- 
tinued a  long  time,  the  apparatus  should  be  such  as  can 
easily  render  the  total  quantity  of  work  expended  by  the 
motor. 

To  meet  the  first  condition,  we  use  plates,  which  bend 
in  proportion  to  efforts  exerted,  and  which  have  the  form 
of  solids  of  equal  resistance.  This  affords  much  assist- 
ance in  making  tabular  statements,  while  it  gives  great 
sensibility  to  the  instrument. 

38.  Eules  for  proportioning  spring-plates. — The  theory 
of  the  resistance  of  materials  to  flexure,  according  with 
the  known  results  of  experiment,  shows  that,  when  a  me- 
tallic plate  of  a  constant  rectangular  section  is  fastened 
at  one  end,  and  subjected  at  the  other  to  the  action  of  an 
effort  P,  perpendicular  to  its  length  or  primitive  direc- 
tion ;  or  when  an  elastic  plate  of  the  same  form  is  placed 
freely  upon  two  supports,  and  subjected  in  the  middle  to 
an  effort  P,  directed  in  the  manner.described,  its  flexure 


DYNAMOMETERS.  35 

F,  so  long  as  it  does  not  exceed  the  limits  of  elasticity, 
will  be  : 

1st.  Proportional  to  the  effort  P  ; 

2d.  Proportional  to  the  cube  of  the  arm  of  the  lever  c 
of  this  effort  ; 

3d.  In  an  inverse  ratio  of  the  width  a  of  the  plate,  in 
a  direction  perpendicular  to  the  plane  of  flexure  ; 

4th.  In  an  inverse  ratio  of  the  cube  of  the  depth  b  of  the 
plate,  at  the  fixed  point  for  the  first  case,  and  at  its  mid- 
dle for  the  second  ; 

5th.  In  an  inverse  ratio  of  a  number  E  constant  for 
each  body,  called  the  coefficient  or  modulus  of  elasticity, 
and  which  expresses  in  pounds  the  weight  required  to 
extend  a  prismatic  bar  of  the  same  material,  with  a  unit 
of  surface  for  its  transverse  section,  to  double  its  primitive 
length,  if  such  change  in  its  dimensions  may  be  made, 
without  varying  the  value  of  E. 

Furthermore,  if  the  longitudinal  profile  of  the  plate 
presents  the  parabolic  form  of  solids  of  equal  resistance, 
the  flexure  will  be  double  that  of  a  plate  of  uniform  thick- 
ness throughout  its  length,  subjected  to  the  same  efforts; 
while  the  resistance  to  rupture  is  the  same  in  both. 

Hence,  we  have  for  springs  of  equal  resistance,  con- 
formably to  theory  and  experience,  the  relation 


a  formula,  by  means  of  which  we  can  calculate  any  one 
of  the  quantities  composing  it  when  the  others  are  known. 
I  have  found  in  the  construction  of  many  spring-plates, 
that  if  made  of  good  German  steel,  properly  tempered  and 
annealed,  the  value  of  the  coefficient  of  elasticity  to  be 
used  will  be 

E  =4273700000  pounds  per  square  foot. 
38.  jRatio  to  be  established  between  the  different  pro- 


36  DYNAMOMETERS. 

portions.  —  The  width  a  of  the  plate  should,  at  most,  not 
exceed  the  limit  of  from  .1312  ft-  to  .164  ft-,  since  the  warp- 
ing produced  by  tempering  increases  with  its  width  and 
creates  difficulties  in  its  adjustment. 

An  examination  of  the  springs  which  I  have  made, 
shows  that  the  flexures  of  springs  remain  proportional  to 
their  efforts,  when  for  the  strongest  they  do  not  exceed  TV 
of  their  length,  and  for  the  weakest  £,  the  measure  being 
taken  from  the  place  where  they  are  embedded. 

With  these  data  it  will  be  easy  to  calculate  the  thick- 
ness I  to  be  given  to  the  plate  at  the  place  of  its  setting, 
so  that  under  a  determinate  effort  it  may  take  a  known 
flexure.  It  is  derived  from  the  following  formula  : 

V-70' 

- 


39.  Longitudinal  Profile  of  the  plates.  —  The  above 
dimensions  being  obtained,  we  determine  the  form  of  the 
longitudinal  profile  by  means  of  the  formula 

s    V 
y=-x 

in  which  5  and  c  being  the  quantities  already  designated, 
A.        e  _  i  #:•  _Z? 


FIG.  15. 

a;  will  represent  the  abscissa  of  the  curve  measured  from 
its  origin  B,  and  y  will  be  the  corresponding  ordinate. 

40.  Disposition  of  the  plates  of  springs.* — The  plates 
of  springs  designed  to  measure  the  traction  of  animal  mo- 
tors upon  wagons,  ploughs,  boats,  &c.,  are  disposed  as 
shown  in  Fig.  16. 

*  For  further  details,  seethe  description  d&Apparrih  dynam&metriques,  etc. 
Chez  L.  Matluas. 


DYNAMOMETERS. 


37 


Two  plates  aa!  and  W  exactly  alike,  with  the  inner 
faces  plane,  and  the  outer  parabolic,  are  terminated   at 


FIG.  16. 


the  ends  by  a  knuckle  joint  of  the  same  width,  pierced 
with  a  drilled  hole.  Small  steel  bolts  traverse  these  holes, 
with  moderate  friction,  and  are  secured  in  the  straps/y 
to  which  they  are  fastened  by  screw  nuts. 

A  posterior  catch  c,  is  pierced  with  an  opening  for 
receiving  the  plate,  which  is  passed  through  it  lengthwise  ; 
a  shoulder  with  its  length  equal  to  the  width  of  the  catch, 
is  prepared  midway  the  plate,  and  fits  this  opening  accu- 
rately. Adjusting  screws  </,  with  conical  points,  hold  the 
plate  in  its  seat. 

The  forward  catch  d  receives  likewise  the  plates  oaf 
and  has  a  ring  r,  to  which  is  attached  the  splinter  bar  or 
rope  upon  which  the  motor  acts. 

The  coupling  of  the  plates  is  for  the  purpose  of  adding 
the  flexures  of  each,  and  increasing  the  sensibility  of  the 
instrument. 

For  the  measure  of  great  efforts  we  may  couple  four 
plates,  whose  resistances  concur  to  form  an  equilibrium 
with  the  power. 


38  DYNAMOMETERS. 

We  guard  against  straining  the  springs  by  fastening 
to  the  posterior  catch  <?,  two  shackles  H9  connected  by 
the  cross  bars  e  e  upon  which  the  forward  catch  impinges, 
when  the  tension  has  reached  the  appointed  maximum 
limit. 

41.  Contrivance  to  obtain  a  permanent  trace  of  the 
flexures  of  the  spring. — The  forward  catch  bar  bears  a 
screw,  through  which  slides,  with  gentle  friction,  a  hollow 
brass  tube  terminated  by  a  conical  socket,  in  which  is 
placed  a  brush  without  the  quill.     The  tube  is  filled  with 
India  ink,  mixed  to  a  proper  consistency. 

When  the  brush  is  well  wet,  and  properly  fixed  in  the 
conical  socket,  the  capillary  attraction  will  afford  a  con- 
stant and  regular  supply. 

We  may  substitute  for  the  brush  a  common  black- 
lead  pencil,  in  which  case  the  tube  and  pencil  should 
weigh  26  pennyweights,  to  make  the  trace  sufficiently 
plain. 

The  traces  of  the  style  are  received  upon  a  band  of 
paper  rolled  upon  a  supply  cylinder  I,  which  passes  over 
three  small  cylinders  guiding  it  under  the  style  and  pre- 
venting any  bending  of  the  paper  from  the  action  of  the 
wind  or  from  its  own  weight. 

The  strip  of  paper  is  rolled  upon  another  barrel  <?, 
serving  as  a  receiver,  upon  which  one  of  its  ends  has  been 
fastened  with  mouth  glue. 

A  second  style  &,  attached  to  one  of  the  shackles,  and 
consequently  immovable,  traces  upon  the  paper  a  line  an- 
swering to  no  effort,  or  to  the  position  of  the  plates  at 
rest,  and  thus  affords  the  zero  of  efforts  ;  so  that  the  effort 
exerted  is  always  measured  by  the  distance  of  the  curve 
traced  by  the  movable  style  from  this  line  of  zero. 

42.  Method  of  moving  the  paper  which  receives  the 
trace  of  the  styles. — The  progressive  motion,  perpendicular 
to  the  direction  of  the  efforts  exerted,  is  transmitted  to 


DYNAMOMETERS.  39 

the  band  of  paper  by  means  of  an  endless  cord,  which 
passes  round  the  hub  of  one  of  the  forward  wheels  of  the 
wagon,  and  over  a  pulley.  Upon  the  prolongation  of  the 
axis  of  this  pulley  is  an  endless  screw,  parallel  to  the 
plates,  and  driving  a  pinion  mounted  upon  the  axis  of  a 
small  cylinder.  Around  this  last  is  wound  a  silk  cord, 
which  transmits  the  motion  to  the  cylinder  receiving  the 
paper. 

By  properly  proportioning  this  transmission  we  may, 
with  a  band  of  paper  from  50  to  59  feet  long,  prolong  the 
experiments  for  distances  of  2,600  to  3,280  feet,  and  even 
greater. 

But  if  the  motion  is  transmitted  directly  to  the  arbor 
of  the  receiving  cylinder,  whose  paper  in  rolling  increases 
the  exterior  diameter,  it  follows  that  while  the  motion  of 
the  cylinder  is  uniform  or  in  a  constant  ratio  with  the 
wheel,  that  of  the  translation  of  the  paper  band  would  be 
accelerated.  To  remedy  this  inconvenience  the  silk  thread 
rolled  upon  the  small  intermediate  cylinder  ^,  is  fixed  at 
its  free  end  to  a  conical  barrel,  whose  surface  channelled 
into  helicoidal  threads,  gives  diameters  calculated  to  com- 
pensate for  the  gradually  increasing  diameter  of  the  re- 
ceiving cylinder. 

43.  Observations  upon  the  quadrature  of  traced  curves. — 
From  this  general  description,  we  see  that  the  paper  un- 
rolling under  the  style,  at  a  velocity  in  constant  ratio 
with  the  distance  run,  the  length  of  the  paper  represents 
this  distance  at  a  scale  known  by  this  ratio. 

The  ordinates  of  the  curves  of  flexure,  measured  from 
the  zero  line,  being  proportional  to  the  efforts  exerted,  it 
follows,  that  the  area  comprised  between  the  curve,  the 
zero  line,  and  any  two  ordinates,  will  represent,  according 
to  what  has  been  said  in  (ISTo.  18),  the  total  work  devel- 
oped by  the  motive  power  in  this  interval. 

44.  Modes  of  operating  this  quadrature. — This  quad- 


DYNAMOMETERS. 


rature  may  be  obtained  by  simple  tracings  and  common 
calculations,  or  by  using  transparent  scales  to  take  an 
abstract  of  the  ordinates :  this  method,  however,  is  tedious, 
and  for  it  we  may  substitute  either  of  the  two  following : 
The  first,  dispensing  with  all  calculation,  consists  in 


Fio.  17. 

laying  down  a  straight  line  AB  parallel  to  the  zero 
line  1O",  at  a  given  distance  from  the  zero  line, 
greater  than,  or  at  least  equal  to,  the  maximum  flexure.  To 
this  ordinate  corresponds  an  assumed  constant  effort, 
equivalent  to  a  known  value  of  work,  and  represented  by 
the  area  of  the  rectangle  MNBA.  Now  abed  ....  NM 
being  the  real  curve  of  efforts  given  by  experiment,  we 
have  the  proportion 

Area  MNBA  :  area  abed  .  .  .  NM  :  :  work  of  assumed 
constant  effort :  work  sought. 

But  the  paper  being  made  by  machinery,  and  of  a 
uniform  thickness,  the  areas  MNBA  and  M#fo  .  .  .  !N"  are 
as  their  weights.  Cutting  them,  therefore,  and  weighing 
first  the  entire  rectangle,  and  then  the  curvilinear  area 
~NLdbc  .  .  .  !N",  we  shall  have  the  work  sought,  by  a  simple 
proportion. 

If,  for  example,  we  have  employed  a  force  of  1543  lb% 
for  which  an  increased  flexure  of  0.0041 ft-  corresponds  to 
an  effort  of  22.05 lbs>,  and  a  constant  flexure  or  rectangular 
height  of  0.2296 ft-  answers  to  an  effort  of  1234lbs- ;  calling 
P  the  weight  of  the  strip  0.2296 ft-  high,  p  the  weight  of 
the  part  contained  between  the  curve  and  the  zero  line, 


DYNAMOMETERS.  41 

E  the  length  of  the  space  described,  and  F  the  mean 
effect  developed  by  the  motor,  we  shall  have 

F=1234.|  pounds, 

and  the  whole  work  of  the  variable  effort  would  have  for 
its  value  the  product  FE. 

45.  Use  of  the  Planimeier. — In  employing  Ernst's 
Planimeter,  furnished  with  a  wooden  cone,  we  have  a 
second  method  of  obtaining  the  quadrature  of  curves  rap- 
idly and  without  calculation. 

This  instrument  is  composed  of  a  cone  l>cb  (Figs.  18 
and  19),  with  its  axis  inclined  to  the  plane  of  the  table 


Plan. 


Fio.  18. 


4:2  DYNAMOMETEES. 

which  supports  the  instrument,  so  that  its  upper  edge  is 
parallel  to  this  plane.  This  cone  rests  by  its  points  upon 
two  supports  fastened  to  the  plate  X,  and  upon  its  pro- 
longed axis  is  a  small  wheel  aa,  which  presses  against  a 
strip  LL  parallel  to  the  guides  along  which  the  plate  XX 
slides;  so  that  when  this  plate  is  pushed  forward  or 
back,  in  the  direction  LL,  the  small  wheel  of  the  cone 
makes  a  number  of  turns  proportional  to  the  space  de- 
scribed by  the  plate. 

A  counter,  the  principal  piece  of  which  is  a  wheel  dd, 
vertical  and  perpendicular  to  the  upper  edge  of  the  cone, 
turning  around  an  axis  parallel  to  this  edge,  is  mounted 
by  points  upon  a  piece  with  slides  ff,  which  is  moved 
with  the  plate  XX,  but  which,  moreover,  may  have  a 
motion  perpendicular  to  the  band  LL,  so  that  the  wheel 


Section  on  A  B. 


Scale  i 


FIG.  19. 


DYNAMOMETERS.  4:3 

can  at  pleasure  be  brought  near  or  removed  from  the 
apex  of  the  cone. 

The  counter  resting  upon  the  surface  of  the  cone  by  its 
own  weight,  when  this  cone  turns  the  wheel  turns  with 
it;  evidently,  the  number  of  turns  it  makes  will  always 
be  proportional :  1st,  to  the  number  of  turns  of  the  cones, 
or  to  the  space  described  in  the  direction  LL ;  and  2d, 
to  the  distance  of  the  wheel  from  the  apex  of  the  cone, 
or  to  the  product  of  these  two  quantities. 

Suppose,  now,  the  wheel  being  at  the  summit  of  the 
cone,  that  a  pointer  g  placed  upon  a  slide  ff^  corresponds 
with  a  line  RS  parallel  to  the  guide  LL,  and  is  over  the 
point  R,  it  is  evident  that  if  we  push  the  plate  XX,  so 
that  this  point  will  follow  exactly  the  line  RS,  the  wheel 
will  not  turn,  since  the  velocity  at  the  summit  of  the  cone 
is  nought ;  but  if  the  point  g  is  at  M,  and  the  wheel  at  a 
distance  from  the  apex  of  the  cone  equal  to  MR=NS, 
when  the  point  is  pushed  from  M  to  N",  the  number  of 
turns  of  the  wheel  will  be  proportional  to  the  length  RS, 
which  is  the  base  of  the  rectangle  MNSR,  and  to  the 
height  of  the  same  rectangle.  It  will  consequently  be  pro- 
portional.to  the  surface  of  this  rectangle.  So  also,  if  we 
cause  the  point  to  follow  the  line  OP,  the  number  of 
turns  of  the  wheel  will  be  proportional  to  the  surface  of 
the  rectangle  ORSP. 

But  in  using  the  instrument  we  cannot  bring  the 
wheel  to  the  summit  of  the  cone,  which  is  truncated,  and 
we  must  somewhat  modify  the  method  of  obtaining  the 
surface  of  the  rectangle  to  be  measured.  Suppose,  for 
example,  it  were  required  to  calculate  the  surface  of  the 
rectangle  OM1STP.  We  first  bring  the  point  g  over  the 
line  MN,  being  careful  that  it  conforms  exactly  to  the 
motion  of  the  plate  XX.  We  then  push  the  instrument 
so  that  the  point  g  shall  pass  from  M  to  !N".  The  counter 
wheel  makes  then  a  number  of  turns,  proportional  to  the 
surface  of  the  rectangle  RMNS.  We  then  draw  the  slide 
ff  and  bring  the  point  g  over  the  point  P,  and  then  draw 


44  DYNAMOMETERS. 

back  the  plate  XX,  so  as  to  have  the  point  g  follow  the 
linePO. 

In  this  retrograde  movement  the  wheel  turns  in  a  con- 
trary direction,  and  makes  a  number  of  turns  proportional 
to  the  surface  of  the  rectangle  ORSP,  and  as  in  these  two 
consecutive  movements  it  has  passed  in  two  opposite 
directions,  it  is  evident  that  the  definite  number  of  turns 
made  is  proportional  to  the  difference  of  the  two  rectan- 
gles OBSP  and  MUSE",  or  to  the  surface  of  the  rectangle 
OMXP. 

The  motion  of  the  wheel  is  transmitted  by  gearing  to 
indicators  with  two  limbs,  one  giving  the  units,  tenths, 
and  hundredths  of  square  millimetres,  and  the  other  the 
thousandths,  the  square  millimetre  being  =.00152  square 
inch. 

What  we  have  said  respecting  the  rectangle  applies 
exactly  to  the  quadrature  of  a  surface  terminated,  as  in  the 
curves  traced  by  the  style  of  dynamometers,  on  one  side 
by  a  straight  line,  and  on  the  other  by  a  curved  line  op, 
for  each  element  of  this  surface  uvxy  may  be  regarded  as 
a  small  rectangle,  whose  base  is  ux,  and  the  height  the 
arithmetical  mean  between  uv  and  xy. 

To  get  the  abstract  of  the  curve,  or  the  quadrature  of 
the  surface  MN/>0,  we  operate  as  follows  :  We  place  the 
sheet  of  paper  under  the  plane  table  of  the  planimeter,  so 
that  the  point  g  being  but  little  removed  from  the  table 
may  follow  exactly  the  line  MK  of  zero  of  efforts,  when 
we  push  the  plate  XX  from  M  to  N.  This  done,  we  bring 
back  the  point  g  over  M,  raise  up  the  counter  and  place 
by  hand  the  indices  at  zero.  We  .then  place  gently  the 
wheel  upon  the  cone,  and  push  the  plate  XX,  so  that  the 
point  g  shall  go  from  M  to  N".  We  then  draw  the  slide 
ff,  so  as  to  bring  the  point  g  over  the  point  p  ;  then,  by 
means  of  the  double  motion  we  can  impart  to  it,  we  fol- 
low exactly  with  this  point  all  the  sinuosities  of  this  curve, 
until  it  arrives  at  o.  We  then  read  upon  the  two  limbs 
the  number  of  square  millimetres  contained  in  the  sur- 


DYNAMOMETEKS. 


face,  and  dividing  it  by  the  length  of  the  base  MN",  ex- 
pressed in  millimetres,  we  have  the  mean  ordinate,  or  the 
height  of  the  rectangle  of  the  same  surface,  which  is  the 
mean  effort  exerted.  But  that  these  operations  j  ust  indi- 
cated should  lead  to  an  exact  result,  we  must  be  sure,  in 
the  forward  or  back  movement,  that  the  wheel  does  not 
slide  along  without  turning.  We  secure  this  result  by 
substituting  for  the  polished  metallic  cone  of  common 
planirneters  one  of  unpolished  wood. 

46.  Dynamometer  for  showing  the  whole  quantity  of 
actions,  for  a  considerable  interval  of  time  and  space. — 
When  we  would  observe  the  work  developed  by  animate 
or  other  motors,  for  a  long  distance,  the  dynamometer 
with  the  style,  whose  band  of  paper  cannot  serve  but 
for  a  distance  of  from  2,000  to  3,000ft-,  will  not  answer  our 
purpose.  Besides,  it  is  quite  often  more  convenient  to 
obtain  at  once  the  quantity  of  work  developed  at  any 
given  distance,  and  it  is  essential  that  we  have  an  appa- 
ratus which  of  itself  shall  record  a  total  of  the  successive 
elementary  quantities  of  work,  and  thus  dispense  with  the 
quadrature,  whose  use  we  have  just  explained.  Such  is 
the  design  of  the  following  modifications  attached  to  the 
dynamometer  described  in  the  preceding  sections  : 

The  posterior  catch  bar 
c  (Fig.  20)  is  traversed  by 
an  axis  of  rotation,  upon 
which  is  screwed  a  plate 
B  with  a  radius  0.26  f% 
placed  above  the  springs, 
and  which  has  at  its  lower 
end  a  pulley  to  which 
the  motion  of  the  wheel 
is  transmitted  by  means 
of  an  endless  cord  passing 
over  pulleys.  A  support 
E  embodied  in  the  ante- 


Qd 
O 

1 L 


FIG.  20. 


46  DYNAMOMETERS. 

rior  catch  bar  d  bears  a  counter,  which  follows  all  the 
motions  of  flexure  of  the  front  spring. 

The  principal  piece  of  the  counter  is  a  wheel  mounted 
upon  an  axis  parallel  to  the  plate  and  to  the  direction  of 
the  efforts  of  traction.  This  wheel  acts  as  that  of  the 
counter  of  the  planimeter,  but,  since  instead  of  the  cone 
we  have  here  a  plane,  it  will  be  at  the  centre  of  the  circle 
when  the  instrument  is  at  rest.  From  what  has  been 
said  in  No.  45,  it  is  needless  to  describe  the  action  of  this 
instrument,  and  we  see  that  the  number  of  turns  of  the 
wheel  is  proportional  to  the  sum  of  elementary  products 
of  the  efforts  exerted,  and  of  the  elements  of  the  path  de- 
scribed, or  to  the  total  work. 

Calling  T  the  distance  in  feet  of  the  wheel  from  the 
centre  of  the  plate,  under  the  effort  of  a  traction  expressed 
in  pounds,  or  the  flexure  of  the  spring  under  this  effort, 
provided  the  instrument  is  arranged  so  that  the  wheel 
rests  upon  the  centre  of  the  plate  when  the  effort  is  zero ; 

r'  the  radius  of  the  small  wheel ; 

e  the  space  described  in  one  second  by  the  wagon  in 
the  direction  of  its  draught,  if  the  effort  is  constant,  and 
in  an  infinitely  small  period,  if  the  effort  is  variable  ; 

E  the  radius  of  the  wheel  from  which  the  motion  is 
derived ; 

p 

71=:——  the  number  of  turns  of  the  wheel  correspond- 

27TXV 

ing  to  the  space  e  ; 
-p 

K=-  the  ratio  of  efforts  to  the  measured  flexures  ; 
T 

!N"  the  number  of  turns  of  the  small  wheel  answering 
to  the  space  e  ; 

R/  the  radius  of  the  hub  of  the  wheel,  by  which  the 
motion  of  the  plate  is  produced ; 

T'  the  radius  of  the  pulley  of  the  plate ; 

It  is  evident  that  this  plate  will  make  a  number  of 
-p/ 

turns  equal  -r  for  one  turn  of  the  wheel,  or,  rather,  to 


DYNAMOMETERS.  47 

T>/ 

— — .—,  for  the  space  e  described  in  the  direction  of  the 

27T.K  T 

draught. 

rip 

The  small  wheel  will  make  —  turns  for  one  turn  of  the 

rl 

plate ;  we  shall  have  then  N=— ?_._.-,,  for  the  number 

of  turns  of  the  small  wheel,  corresponding  to  a  space  £, 
described  under  the  effort  of  a  traction  F. 

The  number  !N"  is  finite  or  infinitely  small,  according 
as  we  deal  with  a  constant  effort,  and  a  finite  space,  or 
with  a  variable  effort,  and  an  element  of  space.  But  we 

TT  TT 

have  by  definition  K=-,  whence  r=^,  and  consequently 

Thus,  whether  for  a  constant  effort  and  a  finite  work, 
or  for  a  variable  effort  and  an  elementary  work,  we  see 
that  the  work  developed  by  the  motor  is  measured  by  the 

product  of  the  constant  factor  -  -^57-—  >  and  of  the  num- 
ber !N"  of  turns,  or  elementary  fractions  of  turns  made  by 
the  small  wheel,  so  that  the  total  work,  at  the  end  of  any 
interval,  being  the  sum  of  the  elementary  quantities  of 
work  successively  developed,  will  be  equal  to  the  same 
product  in  taking  the  number  N  equal  to  the  total  number 
of  turns  of  the  small  wheel  during  the  observed  interval. 
Instruments  of  this  kind  have  been  successfully  em- 
ployed, and  have  afforded  great  facilities  in  prolonged 
experiments  upon  the  draught  of  carriages,  and  have  ena- 
bled us  to  determine  the  total  quantities  of  work  devel- 
oped by  six  horse  teams,  during  their  entire  daily  trips,  and 
for  routes  from  Paris  to  Amiens,  and  from  Nantz  to  Mans. 

47.  Arrangement  to  obtain  the  indications  of  the  num- 
ber of  turns  made  by  the  small  wheel. — We  may  easily 
conceive,  that  when  the  axis  of  the  wheel  has  an  endless 


48  DYNAMOMETERS. 

screw,  its  motion  may  be  easily  communicated  by  properly 
proportioned  gearing  to  two  limbs,  one  of  which  will  give 
the  units  and  tenths  of  turns,  and  the  other  the  hundredths 
and  thousandths  of  turns  of  the  small  wheel.  But  further, 
in  order  to  be  able  to  observe  the  divisions  of  these  limbs 
without  stopping  the  instrument  or  the  trip,  two  styles  are 
so  arranged  that  traversing  two  cups  filled  with  thick  ink, 
they  may  deposit  upon  the  enamelled  limbs  a  black  dot, 
by  placing  the  finger  upon  a  button.  Observations  can 
thus  be  made  and  multiplied,  without  confounding  the 
results. 

48.  Dynamometer  with  chronometer  motors. — When  we 
wish  to  experiment  upon  the  resistance  of  tow-boats  and 
foot-swing  ploughs,  it  would  be  at  least  difficult,  and  in 
some  cases  impossible,  to  put  the  motion  of  the  paper  in 
constant  ratio  with  the  space  described.  In  this  case,  it 
is  very  convenient  to  employ  a  chronometer  motor,  which 
communicates  to  the  paper  a  uniform  motion.  Then  the 
developed  lengths  of  the  paper  represent  the  time,  and 
the  quadrature  of  the  curve  of  flexures  gives  the  sum  of 
the  product  F  x  t  of  each  effort  by  its  elementary  dura- 
tion, or  what  we  call,  as  we  shall  see  hereafter,  the  total 
quantity  of  motion  developed  in  the  observed  interval  of 
time,  or  by  the  length  of  the  developed  paper,  we  have 
the  mean  effort  of  the  motive  power. 

In  the  towing  of  boats,  and  in  all  cases  where  the  ve- 
locity influences  the  results,  we  provide  two  auxiliary 
brushes,  one  of  which  serves  to  mark  upon  the  paper 
intervals  of  time,  15",  30",  &c.,  and  the  other  the  dis- 
tances described  -in  the  passing  of  mile  posts,  or  of  objects 
whose  distances  are  known. 

49.  Rotating  dynamometer. — The  instruments  we  have 
been  describing  are  constructed  for  measuring  the  effort 
or  work  developed  by  motors  whose  action  takes  place  in 
straight  or  circular  lines,  but  it  is  easy  to  modify  them  so 


DYNAMOMETERS. 


as  to  obtain  the  work  transmitted  by  an  axis  of  rotation, 
to  any  machine,  in  applying  the  principle  of  styles,  or 
that  of  the  counters. 

50.  Description  of  a  rotating  dynamometer  with 
styles. — Upon  a  shaft  resting  on  two  cast  iron  supports 
fastened  to  a  wooden  platform,  are  placed  three  pulleys 
of  the  same  diameter  (figs.  21  and  22) ;  the  one  A  is 


FIG.  21. 


Fia.  22. 


50  DYNAMOMETERS. 

fixed,  the  other  0,  near  the  first,  is  loose,  and  the  last,  B, 
is  movable  around  the  shaft,  between  limits  which  we 
shall  indicate. 

This  apparatus  being  placed  between  the  motor  shaft 
and  a  machine  whose  resistance  is  to  be  measured,  the 
loose  pulley  C  receives  the  transmission  belt  of  the  motor 
shaft,  and  when  this  belt  is  passed  over  the  fixed  pulley 
A,  the  shaft  is  set  in  motion  and  acquires  a  velocity  de- 
pending upon  the  ratio  of  the  diameter  of  the  pulley  to 
that  of  the  drum  of  the  motor  shaft. 

The  pulley  B  receives  a  belt  which  serves  to  transmit 
motion  to  the  machine,  and  to  overcome  its  resistance, 
and  as  it  has  but  a  slight  friction  upon  the  shaft,  it  would 
not  be  impressed  with  the  motion  imparted  to  the  shaft 
by  the  fixed  pulley,  unless  a  stop  embodied  in  it  were 
pressed  by  the  extremity  of  a  spring-plate  planted  upon 
the  shaft  in  the  direction  of  one  of  its  spokes.  This  spring 
turning  with  the  shaft,  acts  upon  the  stop,  whose  resist- 
ance bends  it,  and  when  the  resistance  to  flexure  is  able 
to  overcome  that  opposed  by  the  machine,  motion  com- 
mences, and  is  thus  found  to  be  transmitted  from  the  mo- 
tor shaft  to  the  machine  experimented  upon,  through  the 
agency  of  a  spring,  whose  flexures  are  the  immediate 
measure  of  the  resistance  to  be  overcome. 

A  style  adjusted  upon  an  arm  of  the  pulley  can  be 
brought  to  any  desired  proximity  with  the  paper,  endowed 
with  a  motion  of  its  own,  in  a  constant  ratio  with  that  of 
the  pulley  or  the  shaft,  and  then  traces  a  curve  of  flexures 
of  the  spring  exactly  in  the  same  manner  as  in  dynamom- 
eters employed  on  wagons. 

Another  style,  immovable  relatively  to  the  first,  traces 
at  the  same  time  a  line  corresponding  to  a  flexure  zero,  or 
in  the  position  occupied  by  the  movable  style  when  the 
effort  is  zero.  This  line  of  zero  will  be  found  in  the  mid- 
dle of  the  width  of  the  paper,  so  that  the  effort  may  be 
measured  in  either  direction. 

The  springs  used  have  a  parabolic  section,  and  may  be 


DYNAMOMETEKS.  51 

multiplied  at  pleasure,  according  to  the  intensity  of  efforts 
to  be  measured  by  the  instrument. 

A  catch  placed  upon  the  shaft  limits  the  displacement 
of  the  pulley,  and  consequently  the  flexure  of  the  springs, 
so  as  to  prevent  their  being  strained  in  case  of  any  con- 
siderable accidental  efforts. 


51.  Transmission  of  the  motion  of  the  shaft  to  the  land 
of  paper. — A  toothed  ring  is  adjusted  with  gentle  friction 
upon  the  shaft,  and  its  helicoidal  teeth-range  is  geared 
with  a  pinion,  whose  axis,  being  in  a  plane  perpendicular 
to  that  of  the  shaft,  does  not  come  in  its  way.     The  axle 
of  this  pinion  has  an  endless  screw  driving  another  pinion 
mounted  upon  the  prolongation  of  the  axle  of  the  small 
cylinder,  on  which  is  rolled  the  silk  which   drives  the 
fusee.     When  we  wish  to  set  the  band  of  paper  in  mo- 
tion, we  make  the  toothed  ring  immovable  by  means  of  a 
stay,  upon  which  a  catch  fastened  to  this  ring  impinges, 
when  it  is  properly  turned.     Then  the  toothed  ring  being 
fixed  in  space,  while  this  pinion  driven  by  the  shaft  rolls 
around  it,  this  pinion  will  acquire  a  relative  motion  which 
is  transmitted  to  the  screw,  to  the  fusee,  and  to  the  band 
of  paper. 

This  apparatus  is  provided  with  a  conical  fusee  to  con- 
trol the  motion  of  the  bobbin  which  carries  the  paper ; 
by  means  of  this  we  compensate  the  relative  increase  in 
the  velocity  of  translation  of  the  band  of  paper,  which, 
without  this  precaution,  would  ensue  from  the  increase  of 
diameter  of  the  motor  wheel,  upon  which  the  succes- 
sive layers  of  paper  accumulate  according  as  the  trace  of 
the  dynamometric  curve  is  effected. 

52.  Results  of  experiments  made  with  the  rotation  dy- 
namometer.— As  examples  of  the  results  derived  from  the 
rotation  dynamometer,  we  will  report  some  that  were 


52 


DYNAMOMETEKS. 


obtained  at  the  saw-mills  and  wheelwright's  machines  of 
the  imperial  coach  establishment  at  Chaillot : 


is 

1 

C  *  * 

JS 

Kind  of  Machine. 

Condition  and  kind 
of  Wood. 

5j 

E 

i  I 

S  g, 

It 

r 

i 

ir 

f 

feet. 

sq.  ft. 

horse  power. 

Ibs. 

Oak  cut  8  years, 

1.164 

7.6803 

2.82 

163944 

Vertical  saw, 
with  one  blade,  " 

Ash    "    2      " 
Soft  Elm  cut  4  years, 
Aspen        "    4     " 

0.970 
1.570 
1.279 

6.4801 
12.5425 
6.6759 

2.45 
4.60 
2.67 

162470 

198682 
150414 

Twisted  Elm  cut  1  year, 
Ash  cut  8  years, 

0.566 

5.8224 
1.5124 

3.48 

2.80 

260040 
312400 

Circular  saw, 

«       8     " 

0.402 

1.6910 

2.375 

176990 

2.03  ft  in 

"       8     " 

0.284 

1.3455 

1.665 

166986 

diameter. 

«        3      « 

0.187 

0.6480 

1.910 

156772 

"       3     " 

0.068 

0.3240 

1.775 

126368 

WHEELWRIGHT  MACHINES. 


Kind  of  Machine. 

Condition  and  kind  of  Wood,  or 
nature  of  Work. 

Mean  work  in 
Horse   Powers. 

For  sawing  felloes, 
Diagonal  cut  saw, 

Elm  cut  2  years, 
Ash    "   3     « 

1.390 
1.225 

Machine  for  spoke  tenons, 

Oak    "    2     " 

0.460 

"   piercing  Felloes, 

Elm  cut  2  years,  holes  for  spokes, 

0.253 

4t                16                  U                         1C 

"           "         pins, 

0.125 

"        "   making  pins, 

Oak,  pins  of  0.118  feet, 

0.390 

"   piercing  iron, 

Holes  0.115  feet, 

0.551 

Blower  forcing  air  upon      -] 

19  fires  making  1296  "I     t 
13            "           1816  1     tu™s 

A                          (t                       -1  KQ/»     ?~                 <* 

2.860 
2.750 

I 

1327  J   minute> 

1.920 

53.  The  counter  of  the  rotation  dynamometer. — The 
movable  pulley  and  the  mounting  of  the  plate  springs  is 
precisely  the  same  as  in  the  dynamometer  with  styles.  A 
bevel-toothed  ring  with  gentle  friction  is  geared  to  a  con- 
ical pinion  whose  axis  stands  at  right  angles  with  that  of 
the  shaft.  The  axis  of  this  pinion  is  terminated  by  an 
endless  screw,  which  drives  a  toothed-wheel,  whose  axis 
parallel  to  that  of  the  shaft,  carries  at  the  other  end  a 
brass  plate,  with  its  plane  perpendicular  to  the  axle.  The 
movable  pulley  carries  a  wheel  counter  similar  to  that 
described  in  No.  46,  which  is  displaced  with  this  pulley 
a  quantity  proportional  to  the  flexure  of  the  springs.  By 


DYNAMOMETEKS. 


53 


endless  screws  we  can  place  the  small  wheel  in  the  centre 
of  the  plate  when  the  machine  is  at  rest.  The  theory 
and  action  of  this  instrument  is  also  analogous  to  those  of 
dynamometers  with  counters  for  wagons. 

This  instrument  can  easily  be  proportioned  so  as  to 
obtain  the  total  quantity  of  work  transmitted  by  a  rotating 
axle,  during  a  day,  a  week,  or  a  month,  and  in  this  regard 
will  be  very  useful  in  observations  relative  to  the  distri- 
bution of  motive  force  among  different  work-shops,  or  the 
consumption  of  fuel  by  steam-engines. 

54.  Gauge  of  pressure 
of  steam  in  the  cylinders  of 
engines. —  Watt's  gauge  per- 
fected ly  Mac-Naught. — 
It  is  of  the  greatest  utility, 
for  appreciating  the  effects 
of  the  distribution  of  steam 
in  the  interior  of  cylinders 
of  steam-engines,  to  have 
the  means  of  measuring  the 
pressure  of  the  steam  at 
different  points  of  the  stroke 
of  the  piston.  Watt  gave 
his  attention  to  the  con- 
struction of  a  small  instru- 
ment for  this  purpose,  which 
he  named  Indicator  of  Pres- 
sure, and  which  since  his 
time  has  received  many 
improvements  in  its  details. 
It  is  composed  of  a  free 
piston,  with  moderate  fric- 
tion, and  without  packing, 
fig.  23,)  contained  in  a  small 
cylinder,  terminated  at  its 
lower  end  by  a  tube,  provid-  Flo>  28. 


54:  DYNAMOMETERS. 

ed  with  a  stop-cock  screwed  on  to  the  head  of  the  cylin- 
der. When  the  cock  is  open,  the  steam  rushing  into  the 
cylinder  tends  to  drive  the  piston  upwards,  but  the  stem 
of  it  being  connected  with  a  spiral  spring,  this  spring  is 
compressed  and  serves  to  measure  the  effort  exerted. 
With  this  kind  of  spring,  and  with  cards,  we  may  ob- 
tain flexures  proportioned  to  the  efforts,  and  need  only 
a  trace  of  these  flexures.  For  this  purpose,  the  stem 
of  the  small  piston  bears  a  pointed  arm  or  lever, 
furnished  with  a  crayon,  which  is  brought  in  contact 
with  a  sheet  of  paper  rolled  upon  a  copper  cylinder, 
whose  axle  is  parallel  to  the  stem  of  the  piston;  a 
groove  is  made  upon  the  lower  part  of  this  cylinder,  in 
which  winds  round  a  thread,  the  end  of  which  is  fastened 
to  a  small  winch.  The  number  of  turns  of  the  thread 
around  this  winch  has  a  development  a  little  less  than 
that  of  the  cylinder,  and  upon  its  axle  is  a  pulley  receiving 
many  turns  of  a  thread,  whose  development  is  equal  or 
superior  to  the  stroke  of  the  piston.  Within  the  cylinder 
is  a  spiral  spring  which  forces  it  back  to  its  first  position 
in  the  return  stroke  of  the  piston. 

It  follows  from  this  disposition,  that  during  the  intro- 
duction and  expansion  of  the  steam,  the  style  will  trace 
upon  the  sheet  of  paper  a  curve  giving  the  excess  of  the 
internal  over  the  external  pressure  ;  then  that,  in  the  pe- 
riod of  its  escape,  the  cylinder  turning  back,  the  style  will 
trace  another  curve  giving  the  pressure  during  the  escape, 
and  that  the  second  branch  on  the  following  stroke  closes 
in  upon  the  first.  The  length  of  the  paper  developed 
being  proportional  to  the  stroke  of  the  piston,  and  the 
ordinates  bounded  by  the  two  'curves  being  in  all  cases 
proportioned  to  the  motive  pressure  of  the  steam,  it  is 
evident  that  the  area  of  surfaces  comprised  within  these 
curves  represent  the  work  developed  upon  the  small  pis- 
ton, and  consequently  that  upon  the  great. 

The  use  and  application  of  this  instrument  is  easy,  and 
may  give  good  indications,  even  though  somewhat  worn, 


"DYNAMOMETKRS. 


55 


but  we  would  remark  that  when  the  crayon  has  traced 
many  successive  curves,  they  become  confounded  or  over- 
lay each  other,  so  as  sometimes  to  create  confusion ;  still, 
the  facility  of  its  establishment  causes  it  to  be  in  great 
demand  with  constructors  of  steam-engines. 

55.  New  style  indicator. — To  avoid  the  confusion  of 
curves,  I  have  proposed  to  adapt  to  the  indicator  the 
arrangement  used  for  common  dynamometers.  (Figs. 
24:,  25,  and  26.)  Instead  of  acting  upon  a  spiral  spring, 


FIG.  25. 


the  piston  of  the  instrument  has  a  square  head  d  pierced 
with  an  opening,  in  which  is  fastened  the  end  of  a  parabolic 
spring  plate,  the  other  end  being  secured  to  a  support  /. 
the  plate  has  such  a  length  that  on  either  side  it  may 
bend  several  centimetres,  (cent. =.0328"-)  and  as  we  may 


56 


DYNAMOMETERS. 


use  plates  more  or  less  rigid,  the  instrument  may  serve  to 
measure  pressures,  comprised  between  one  and  ten  atmos- 
pheres. Thus,  for  example,  for  a  high-pressure  engine 
working  at  four  atmospheres  above  the  atmospheric  pres- 
sure, each  atmosphere  may  correspond  with  a  flexure  of 
from  .0328 ft-  to  .0361 ft-  of  the  spring,  which  is  exact 
enough  in  practice.  The  head  of  the  piston  carries  in 
front  of  the  spring  blade  a  style  g,  which  traces  upon  a 
sheet  of  paper  the  curve  of  flexures,  or  the  tensions  of 


FIG.  26. 

the  steam.  Another  fixed  style  A,  adjusted  so  as  to  trace 
the  same  right  line  with  the  movable  style  when  the  spring 
is  at  rest,  indicates  the  zero  of  pressures.  When  the 
steam  is  let  in  upon  the  piston  of  the  machine,  it  drives 
that  of  the  instrument  outward,  and  the  curve  traced  is 
beyond  the  line  of  zero :  when,  on  the  other  hand,  the 


DYNAMOMETERS.  57 

steam  expands  and  escapes,  whether  in  the  air  or  the  con- 
denser, the  curve  approaches  the  line  of  zero,  and  may 
pass  by  it.  In  either  case  we  have  upon  the  band  of 
paper  a  trace  of  all  the  variations  of  pressure. 

A  third  fixed  style  k  marks  at  each  stroke  a  point 
which  serves  to  connect  the  curves  with  the  commence- 
ment of  the  stroke  of  the  piston.  Notwithstanding  the 
advantages  possessed  by  this  instrument,  for  the  study  of 
the  effects  of  steam-engines,  by  the  multiplicity  and  dis- 
junction of  its  curves,  we  must  admit,  that  for  common 
use,  "Watt's  improved  indicator,  in  its  greater  portability, 
and  convenience  of  establishment,  answers  quite  as  well 
for  ascertaining  the  condition  of  a  steam-engine. 

We  see  that  the  two  principles  upon  which  are  founded 
all  the  instruments  we  have  described,  to  wit :  1st.  The 
use  of  a  style  tracing  a  curve  of  efforts  upon  a  sheet  of 
paper  set  in  motion  by  direct  means ;  and  2d.  The  use  of 
a  small  wheel  counter,  to  totalize  the  quantity  of  work, 
readily  applies  to  All  kinds  of  observations  we  may  have 
to  make ;  and  finally,  I  would  bear  in  mind  that  the  main 
idea  of  these  two  solutions  of  the  questions  we  have  dis- 
cussed, were  pointed  out  to  me  by  M.  Poncelet,  my  friend 
and  teacher,  and  whatever  I  may  claim  in  the  construc- 
tion of  these  instruments  is  only  relative  to  the  realization 
of  this  pregnant  and  ingenious  thought. 


THE  TRANSMISSION  OF  MOTION  BY  FORCES. 

56.  General  remark  relative  to  the  laws  of  motion. — 
We  have  derived  from  mechanical  geometry  a  knowledge 
of  the  laws  of  uniform  motion,  as  of  those  of  motion  uni- 
formly accelerated  or  retarded.  Experience  also  shows 
us,  that  there  exists  motions  subjected  to  these  laws. 
Thus,  for  example,  we  admit,  by  means  of  various  chro- 
nometric  contrivances  adapted  to  these  observations,  that 
the  motion  of  descent  of  different  formed  bodies  in  air  or 
water,  quickly  becomes  uniform  when  they  present  sur- 
faces sufficient  for  the  resistance  of  tl^e  air  to  acquire  a 
suitable  intensity. 

We  also  admit  that  heavy  bodies  with  small  surfaces 
fall  to  the  earth  with  a  uniformly  accelerated  velocity. 

These  facts  established,  it  is  proper  to  deduce  their 
consequences. 

We  know,  (No.  3)  according  to  the  fundamental  prop- 
erty of  matter  called  inertia,  "  that  all  bodies  in  a  state 
of  uniform  motion  proceed  in  the  same  straight  line, 
unless  some  obstacle  constrains  them  to  change  that 
state." 

If,  then,  a  body  is  impressed  with  a  uniform  motion 
so  that  no  foreign  cause  or  force  operates  to  change  this 
state  of  motion,  or  if  many  forces  solicit  it  to  equal 
changes,  their  action  will  be  counterbalanced,  neutralized, 
and  will  be  in  equilibrium. 

Such  is  the  case  with  parachutes  descending  with  a 
uniform  motion.  The  action  of  gravity,  and  that  of  the 
resistance  of  the  air,  compensate  and  destroy  each  other. 


TRANSMISSION  OF  MOTION   BY  FORCES.  59 

57.  Consequences  relative  to  the  causes  producing  accel- 
erated or  retarded  motion. — In  motion  uniformly  acceler- 
ated or  retarded,  the  increase  or  diminution  of  velocity 
being  always  the  same  for  equal  times,  the  force  producing 
this  modification  of  motion  is  then  constant,  since  it  pro- 
duces constant  effects.    Thus,  when  observation  has  shown 
us  that  the  motion  is  uniformly  accelerated  or  retarded, 
we  are  justified  in  the  conclusion,  that  the  force  which 
accelerates  or  retards  is  constant. 

58.  Vertical   motion   of   heavy  bodies. — Experiment 
proves  that  in  a  vacuum,  all  bodies  subjected  to  the  action 
of  gravity  fall  from  the  same  height  in  the  same  time, 
whatever  their  density.     It  follows  from  this,  that  gravity 
operates  in  the  same  manner  upon  all  the  material  mole- 
cules.    In  air  and  other  resisting  mediums,  the  resistance 
experienced  by  bodies  depends  on  the  extent  and  form  of 
their  surfaces,  and  the  nature  of  their  motion  is  notably 
modified,  when  the  velocities  are  great,  and  the  bodies 
have  very  great  bulk  relative  to  their  weight.     But  for 
bodies  such  as  stones,  wood,  metal,  used  in  construction 
and  for  common  heights  of  fall,  the  resistance  of  the  air 
is  so  small,  that  we  may  usually  leave  it  out  of  account. 

Galileo,  in  observing  the  times  employed  by  bodies 
rolling  upon  inclined  planes  or  falling  vertically,  was  the 
first  to  observe  the  fact,  that  the  spaces  described  in  a 
vertical  direction,  and  in  that  along  planes,  were  to  each 
other  as  the  squares  of  the  times  employed ;  whence  he 
concluded  that,  "for  the  same  place  upon  the  surface  of 
the  earth  gravity  was  uniform  and  constant"  It  is  thus 
that,  from  experiment,  has  been  derived  this  important 
mechanical  law.  Applying  to  this  case  the  laws  which 
we  have  found  for  all  accelerated  or  retarded  motions,  we 
shall  have  for  the  velocity  imparted  or  destroyed  at  the 
first  second,  and  which  is  usually  designated  by  the  letter 
^,  Vt  =#=32.1817 ft-  The  space  described  in  the  vertical 


60  TRANSMISSION   OF  MOTION   BY  FORCES. 

direction,  or  the  height  is  designated  by  the  letter  H. 
We  have  then  for  the  formula  of  motion  of  heavy  bodies 


TT 


59.  Use  of  this  formula. — The  first  formula  serves  to 
determine  approximately  the  height  of  a  tower,  or  depth 
of  a  pit,  by  a  simple  observation  of  the  duration  of  the 
fall  of  a  body.  If,  for  example,  we  have  found  that  a 
body  (for  which,  in  case  we  try  a  pit,  we  make  use  of  a 
light)  has  taken  2.5"  to  pass  from  the  curb  to  the  bottom 
of  a  pit,  we  shall  have  for  its  depth 

11=16.0908  x  (2.5//)2=100.57fu 

The  third  is  of  frequent  use,  especially  in  calculating 
the  gauging  of  the  discharge  of  water,  and  gives  the 
velocity  corresponding  to  a  known  height. 

Thus,  for  a  height  H=3.9371ft-  we  find 


Y=  1/64363  x3.9371=15.91ft- 

It  has  been  reduced  in  tables,  which  may  be  found  in 
most  of  the  works  on  mechanics ;  but  the  rule  for  calcula- 
tion is  a  substitute  for  these  tables,  when  they  are  not  at 
hand.  Bringing  a  pointer  under  the  number  64.363,  read 
at  the  upper  scale,  we  find  in  the  lower  scale  the  veloci- 
ties corresponding  to  all  the  heights  read  upon  the  reglet, 
or,  reciprocally,  reading  the  velocities  at  the  lower  scale, 
we  find  upon  the  reglet  their  corresponding  heights. 

60.  Successive  fall  of  heavy  lodies. — The  laws  of  the 
motion  of  descent  of  heavy  bodies  serves  to  explain, 
among  other  phenomena,  that  of  the  increasing  separation 


TRANSMISSION   OF  MOTION   BY  FORCES.  61 

of  bodies  ;  of  water-drops,  for  example,  which  raised 
together  and  contiguously  in  a  jet  of  water,  fall  in  a 
shower  of  separate  drops.  In  fact,  it  is  easy  to  see  that 
the  drops  starting  from  the  summit  of  the  curve,  one  after 
the  other,  must  separate  more  and  more.  Suppose,  for 
instance,  that  a  drop  of  water  commences  its  descent  0.01" 
before  the  following :  V  after  the  starting  of  the  second, 
the  first  drop  will  have  fallen  during  1.01",  and  through 
a  height 

H=16.0908ft-  x  (1.01)2=16.4:lffc- 

while  the  next,  which  has  been  only  1"  in  motion,  will 
have  fallen  only 

H=16.0908xl'/2=16.091ft- 

Already  the  first  is  in  advance  of  the  second  by  0.31ft-  and 
the  separation  constantly  increasing,  the  jet  falls  back  in 
rain. 

61.  Principle  of  the  proportionality  of  forces  to  their 
velocities. — The  observation  of  facts  shows,  and  it  seems 
quite  natural  to  admit,  ih&t  forces  are  really  proportional 
to  the  degrees  of  velocity  which  they  impress  in  equal  infi- 
nitely small  times,  upon  the  same  ~body,  yielding  freely  to 
their  action  and  in  the  proper  direction  of  this  action. 
This  is  one  of  the  fundamental  axioms  admitted  by  all 
geometricians,  and  is  proved  in  the  exactitude  of  conse- 
quences deduced  from  it. 

If,  then,  we  call  F  and  F'  two  forces  which,  acting 
successively  upon  the  same  body,  impress  it  with  or  de- 
prive it  of  infinitely  small  degrees  of  velocity,  v  and  v'9  in 
an  element  of  time  £,  we  shall  have  from  this  principle 
the  proportion 

F  :  F  :  :  v  :  vr. 

To  get  the  expression  and  measure  of  the  force  F,  we 
may  compare  it  with  another  force,  whose  effect  upon  the 


62  TRANSMISSION   OF  MOTION  BY  FORCES. 

body  is  known  ;  with  gravity,  for  example,  and  as  we  know 
that  the  velocity  imparted  to  heavy  bodies  in  an  element 
of  time  is  vf=gt^  and  as  we  designate  by  P,  the  weight  of 
the  body,  or  the  force  exerted  by  gravity,  the  above  pro- 
portion will  then  become 

F  :  P  :  :  v  :  gt  ; 
Pv 


Before  proceeding  farther,  we  remark  that  the  same 
principle  applied  to  actions  exerted  by  gravity  upon  the 
same  body  in  different  places,  where  the  weight  of  the 
body  is  respectively  P  and  P',  gives  us  the  proportion 


P  :  P'  :  :  gt  :  g't  :  :  g  :  g'  , 

p 

— 
9 


p    p' 

whence  it  follows  that  the  ratio  —  —  —  r  is  constant  for  all 


places  upon  the  earth. 

This  constant  ratio  of  the  weight  of  a  body  to  the 
velocity  communicated  to  it  by  gravity,  in  the  first  second 
of  its  action,  is  what  we  term  its  mass,  and  is  designated 
by  the  letter  M. 

62.  The  measure  of  motive  forces  and  of  inertia.  —  "We 
have,  then,  for  the  expression  of  the  force  F  capable  of 
imparting  to  or  taking  from  a  body  of  the  weight  P  or 
mass  M  an  element  of  velocity  v,  in  an  element  of  time  t 


We  see,  by  this  expression,  that  when  the  weight  of  a 
body  is  given,  or  its  mass,  we  shall  have  the  value  and 
measure  of  its  force  in  pounds,  when  we  know  the  ratio 

-.     If,  for  example,  this  ratio  is  constant,  which  is  the 
t 


TRANSMISSION   OF  MOTION   BY   FORCES.  63 

case  with  motion  uniformly  accelerated  or  retarded,  the 
force  F  is  constant. 

But,  since  to  communicate  to  a  body  of  the  weight  P, 
a  variation  of  velocity  v9  in  an  element  of  time  £,  there 

Pv 

must  be  developed  an  effort  — ,  then  there  is  a  resistance 

gt 

to  be  surmounted,  of  which  this  effort  is  the  measure. 

This  resistance  is  the  force  of  inertia,  the  reaction 
which  takes  place  every  time  that  a  variation  of  motion 
is  produced.  Thus  the  preceding  expression  will  be  at 
once  the  measure  of  the  motive  force,  which  produces  the 
change  of  motion  and  that  of  the  force  with  which  the 
body,  by  virtue  of  its  inertia,  opposes  or  resists  this  change. 

Yv 

An  examination  of  the  formula  F— —   shows  that,  for 

gt 

a  weight  P,  or  a  given  mass  M,  the  magnitude  of  the 
force  F  will  increase  as  the  change  of  motion  becomes 

rt* 

more  rapid,  or  the  ratio  -  becomes  greater.     It  is  thus  we 

t 

account  for  the  magnitude  of  efforts  and  reactions  devel- 
oped in  the  transmission  of  motion  by  the  shocks  expe 
rienced  between  hard  bodies,  in  very  short  intervals  of 
time,  when  the  velocity  varies  or  is  destroyed  stiddenty. 

r\\ 

This  ratio  -  of  the  increase  or  diminution  of  velocity 
t 

in  the  element  of  time  during  which  this  change  is  pro- 
duced, is  that  to  which  for  many  years  past  geometricians 
have  given  the  name  of  acceleration. 

Thus,  in  treating  upon  the  action  of  gravity,  the  con- 
stant acceleration  produced  by  it  is  represented  by  the 

number  g=-. 
t 

It  follows  from  this  definition  and  the  preceding  gen- 
eral principles,  that  the  force  which  produces  an  elemen- 
tary change  in  the  motion  of  a  body,  is  proportional  to 


64:  TRANSMISSION   OF  MOTION  BY  FORCES. 

the  weight  P,  or  to  its  mass  -,  and  to  the  acceleration  - 

9  * 

which  it  produces. 

"We  may  make  sensible  the  increase  of  effort  F  to  be 
exerted,  with  the  rapidity  of  the  communication  of  mo- 
tion, by  means  of  a  spring  balance,  or  any  kind  of  spring 
whose  flexure,  indicated  by  a  style  or  a  scale,  is  so  much 
the  greater  as  the  transmission  of  motion  is  more  rapid. 
If,  for  example,  we  suspend  to  a  spring  balance  a  weight 
of  10lb%  in  which  case  a  pasteboard  scale  placed  opposite 
the  upper  part  may  stop  at  the  fifth  division,  and  then 
raise  the  balance  and  weight  with  an  accelerated  motion, 
the  spring  will  bend  still  more,  and  so  much  the  more,  as 
the  acceleration  of  motion  is  the  more  rapid.  The  in- 
crease of  flexure  indicated  by  the  displacement  of  the 
scale  will  measure  the  effort,  the  resistance  opposed  by 
inertia  to  the  acceleration  of  motion. 

63.  Case  when  the  force  is  constant. — If  the  force  F,  or 
the  ratio  j  is  constant,  we  have  then  at  the  end  of  a  cer- 

5 

tain  time  T,  when  the  force  has  communicated  or  destroyed 
a  velocity  Y,  the  equality 

7=™ ,  and  consequently  F=M— =M- ; 
o     J.  JL         t 

whence 

FT=MY  and  Ft=M.v. 

64.  delation  offerees  to  accelerations. — If  two  forces 
F  and  F'  act  in  succession  upon  the  same  body,  and  im- 

n\  ny 

part  to  it  different  accelerations  -  and  - ,  we  see  that  they 

t          t 

will  be  proportional  to  these  accelerations,  and  that  we 
shall  have 

F  :  F  •  •  2  •  v- 
t'f 


TRANSMISSION   OF  MOTION   BY   FORCES.  65 

It  is  by  reason  of  this  proportionality  that  the  acceler- 
ations are  sometimes  taken  for  the  measure  of  forces. 
But  these  quantities  cannot  be  an  exact  measure  of  forces, 
inasmuch  as  they  only  express  a  ratio. 

Thus,  when  we  say  absolutely  and  without  other  ex- 
planation that  the  quantity  ^,  which  expresses  the  accel- 
eration produced  by  gravity,  is  the  measure  of  this  force, 
we  give  to  students  an  incorrect  idea,  since  g  is  in  reality 
only  the  velocity  imparted  to  or  taken  from  a  body  by 
gravity  during  each  second  of  its  action,  and  the  velocity 
which  is  expressed  in  feet  cannot  measure  a  force  which 
should  be  compared  with  pounds. 

65.    Quantity  of  motion. — The  products  MY,  Mv, 

P  P 

equal  to  — Y  or  —v,  have  received  the  name  of  quantity 

&  \J 

of  motion :  it  is  a  conventional  phrase  to  which  we  attach 
no  other  signification  than  that  of  the  product  of  a  mass 
into  the  velocity  imparted  to  or  taken  from  it. 

"We  would  further  observe,  that  this  product  MY,  M-y, 
is  equal  to  FT  or  F£,  of  the  force  and  time  during  which 
it  has  acted.  If  we  consider  two  forces  as  acting  for  dif- 
ferent times  upon  two  bodies  of  unequal  mass,  we  shall 
have 

F$=M*>,  FY=MV; 

and  consequently 

F* :  FY  :  :  Mu  :  M V ; 

whence  it  follows  that  the  quantities  of  motion  M-y,  MV, 
imparted  to  or  taken  from  different  bodies  in  unequal  times, 
are  as  the  product  of  the  forces  to  which  they  are  due, 
into  the  time  during  which  these  forces  have  acted. 

It  is  only  when  the  times  are  equal  that  the  quantities 
of  motion  impressed  or  destroyed  are  proportional  to  the 
forces,  and  can  serve  for  their  measure. 

From  the  preceding  remarks  it  follows,  as  we  shall 
explain  in  the  following  section,  that  in  shocks  there  is  no 
5 


66  TRANSMISSION  OF   MOTION   BY   FORCES. 

loss  of  quantity  of  motion,  which  is  expressed  in  saying 
that  there  is  a  preservation  of  the  quantities  of  motion. 
But  we  shall  see  hereafter  that  shocks  occasion  a  loss  of 
work. 

66.  Equal  forces  acting  during  equal  times. — If  the 
forces  are  equal  and  act  during  the  same  time,  the  quan- 
tities of  motion  imparted  or  destroyed  in  the  two  bodies 
with  masses  M  and  M'  are  equal.  This  occurs  in  the  re- 
action of  two  bodies  which  press,  push,  or  impinge  upon 
each  other.  The  efforts  of  compression  and  resistance 
being  equal,  opposed  and  developed  during  the  same  time, 
it  follows  that  the  quantity  of  motion  imparted  in  the 
reaction,  to  one  of  the  bodies,  is  equal  to  that  which  is  lost 
by  the  other.  Here  is  a  fact  which  is  a  necessary  con- 
sequence of  the  theory  of  the  shocks  of  bodies. 

Thus,  for  example,  when  a  body  with  a  mass  M  im- 
pressed with  a  velocity  V,  impinges  on  a  body  with  a 
mass  Mx  animated  with  a  velocity  V,  in  the  same  line, 
whether  in  the  same  or  opposite  directions,  it  develops  at 
the  point  of  contact  equal  and  opposite  efforts  of  com- 
pression, in  an  element  of  time  £,  taking  from  the  imping- 
ing body  -a  small  degree  of  velocity  -y,  and  consequently 
a  quantity  of  motion  Mv,  and  imparting  to  the  body 
shocked,  if  it  moves  in  the  same  direction  as  the  first, 
an  increase  of  velocity  v'  and  a  quantity  of  motion  MV. 
These  quantities  being  equal,  we  have  then,  at  each  instant 
of  the  mutual  shock  or  compression  of  bodies,  M.v=~M.fv'. 

In  this  case  one  of  the  bodies  loses  a  quantity  of  motion 
equal  to  that  gained  by  the  other,  and  the  sum  of  their 
two  quantities  of  motion  remains  the  same. 

The  same  thing  transpiring  at  each  instant  of  the 
shock,  it  follows  then  that  the  total  quantity  of  motion 
lost  by  a  body  is  equal  to  that  gained  by  the  other  during 
the  compression,  and  that  at  each  end  of  this  period,  the 
sum  of  their  quantities  of  motion  is  the  same  after  the  shock 
as  before.  This  consequence  constitutes  the  principle  of 


TRANSMISSION  OF  MOTION  BY  FORCES.  67 

the  conservation  of  the  quantities  of  motion,  otherwise 
termed  the  principle  of  the  conservation  of  motion  of  the 
centre  of  gravity. 

If  we  are  dealing  with  soft  bodies,  whose  elasticity  is 
completely  impaired  by  the  shock,  and  which  after  com- 
pression unite  and  travel  together  with  a  common  velocity 
U,  the  quantity  of  motion  after  the  shock  is  (M+M)TJ, 
and  from  what  proceeds  we  should  have 


whence  we  derive  for  the  common  velocity  after  the  shock 

TT_MY+M/Y/ 

M+M~ 

If  the  body  shocked  was  at  rest,  we  should  have  V=0, 
and  the  above  expression  is  reduced  to 

MV 


U= 


M+M'' 


If,  in  the  first  of  these  two  expressions,  we  divide  the 
two  terms  of  the  fraction  by  the  mass  M'  of  the  body 
shocked,  the  common  velocity  after  the  shock  becomes 

-V+V 
M' 


Under  this  form  we  see  that  the  common  velocity  of 
motion  of  two  soft  bodies  will  differ  so  much  the  less  from 
the  velocity  V  of  the  body  shocked  as  the  mass  M  of  the 
impinging  body  is  smaller  compared  with  the  body 
shocked.  At  the  limit,  or  when  the  impinging  body  is 
infinitely  small  compared  with  the  body  shocked,  the  ra- 

tio ^  vanishes,  and  we  have  U=  V,  that  is  to  say,  the 
velocity  of  the  mass  shocked  will  not  be  changed.    This 


68  TRANSMISSION   OF  MOTION  BY   FORCES. 

case  occurs  in  the  motion  of  liquids  and  elastic  fluids, 
when  infinitely  thin  edges  impinge  successively  upon  finite 
masses  endowed  with  a  less  velocity  in  the  same  direction. 
If  bodies  strike  against  each  other  in  opposite  direc- 
tions, a  similarity  of  action  exists  ;  but  then,  at  the  end 
of  the  compression,  either  the  bodies  are  both  brought  to 
a  state  of  rest,  and  we  have 

MV=M'V  and  U=O, 

or  one  of  the  two  goes  backwards,  and  they  proceed  with 
a  common  velocity  U.  If  it  is,  for  example,  the  body  M' 
which  goes  backwards,  the  quantity  of  motion  lost  by  the 
body  M  is  M(V—  U),  and  the  quantity  of  motion  devel- 
oped during  the  period  of  compression,  by  the  forces  of 
reaction  upon  the  body  M',  is  composed  of  that  which 
has  been  destroyed,  or  M'V,  plus  that  imparted  in  an  op- 
posite direction  M'U,  and  since  the  quantities  of  motion 
developed  on  both  sides  upon  each  of  the  bodies  should 
be  equal,  we  have 


whence  we  deduce  for  the  common  velocity,  after  the 
shock  or  compression, 


M+M' 

a  formula  in  which  we  also  see  that  the  velocity  of  the 
impinging  body  will  be  so  much  the  less  changed,  as  its 
mass  M  is  greater  in  its  ratio  with  that  of  the  body 
shocked — for,  dividing  both  parts  of  the  fraction  by  the 
mass  M  of  the  impinging  body,  we  have 


TRANSMISSION  OF  MOTION  BY   FORCES.  69 

This  shows  that  in  machines  working  by  shocks  we 
must  increase  the  weight,  the  mass  of  the  impinging 
pieces,  in  their  ratio  to  the  pieces  shocked,  in  a  ratio  so  much 
the  greater,  as  it  is  desired  to  maintain  a  greater  regularity 
of  motion. 

If  the  body  shocked  is  at  rest,  such  as  a  pile  driven  by 
a  ram,  we  have  V=O,  the  common  velocity  of  the  de- 
scent of  the  pile  and  ram  after  the  shock  is 


_ 

,M/ 

+ 


Which  shows  that  this  velocity  will  differ  so  much  the 
less  from  that  of  the  arrival  of  the  ram  upon  the  head  of 
the  pile,  as  the  mass  M  of  the  rani  is  greater  in  its  ratio 
with  that  of  the  pile. 

It  is  best  in  this  case,  then,  to  increase  the  mass  of  the 
ram  rather  than  its  velocity,  for  the  work  employed  to 
raise  it  increases  only  with  its  weight,  while  its  work  will 
be  increased  proportionally  to  the  height  of  elevation,  or 
to  the  square  of  velocity  of  its  descent. 

67.  Proof  of  the  preceding  considerations  ly  direct 
experiment.  —  The  results  which  we  have  recorded  relative 
to  the  shock  of  soft  bodies  have  been  verified  by  direct 
experiments,  made  by  me  at  Metz  in  1833,*  with  the  fol- 
lowing apparatus  :  A  wooden  box  (Fig.  27),  in  which  was 
placed  successively  clay,  more  or  less  soft,  sand,  pieces  of 
wood,  &c.,  was  suspended  to  a  dynamometer  having  a 
style  and  turning  plate.  The  plate  was  impressed  with  a 
uniform  motion,  which  was  transmitted  by  a  weight,  and 
regulated  by  a  fan  fly-wheel.  "When  the  box  was  im- 
movable, the  resistance  of  the  dynamometer  was  in  equi- 

*  New  experiments  upon  friction,  and  upon  the  transmission  of  motion  by 
shock,  <fec.,  made  at  Metz  in  1833,  by  Arthur  Morin,  Captain  of  Artillery. 


70 


TRANSMISSION   OF   MOTION   BY   FOECES. 


librium  with  the  weight,  and  the  curve  of  flexure  traced 
by  the  style  upon  the  plate  was  a  circle.     The  impinging 


Fio.  27. 

body  was  a  cannon  ball  held  by  tongs,  opening  at  pleas- 
ure, and  when  it  struck  the  materials  in  the  box,  it  caused 
compression,  immediately  after  which  the  two  bodies  fell 
together  with  a  common  velocity.  The  amplitudes  of  this 
motion  were  measured  and  indicated  by  the  flexure  of 
springs,  and  the  result  of  this  was  a  curve  upon  the  plate, 
whose  distance  from  the  axis,  or  radius  vector,  went  on 


TRANSMISSION   OF  MOTION   BY   FORCES. 


71 


increasing  during  all  the  period  of  compression,  or  of  ac- 
celerated motion,  whence    it  followed  that    the   curve 


FIG.  28. 

t 

became  at  once  convex  to  the  circle  of  repose.  Then 
starting  from  the  instant  when  the  compression  had 
attained  its  maximum,  the  bodies  being  soft  or  nearly  so, 
it  followed  that  all  solicitations  of  an  increasing  effort 
upon  the  box  being  discontinued,  the  reaction  of  the  spring 
begins  to  slacken  the  motion  of  descent,  stops  it,  then 
raises  up  the  box  above  its  initial  position,  and  then  causes 
it  to  continue  a  series  of  vertical  oscillations,  which  are 
finally  stopped  by  the  passive  resistances  of  the  apparatus. 

The  abstract  of  these  curves,  and  their  transformation 
into  other  curves  whose  abscissa  are  the  times  propor- 
tional to  the  angles  described,  and  whose  ordinates  are 
the  vertical  spaces  described  by  the  box,  is  quite  an  easy 
matter,  and  are  presented  by  the  figure  itself. 

The  curve  of  motion  being  at  first  convex,  then  con- 
cave towards  the  axis  of  abscissa,  showing  that  the  mo- 
tion was  at  first  accelerated,  then  retarded,  (Nos.  11  and  12,) 


TRANSMISSION   OF  MOTION  BY  FOECES. 


FIG.  29. 


it  is  furthermore  evident  that 
the  velocity  which  is  given  by 
the  inclination  of  the  tangent 
upon  the  axis  of  abscissa,  at- 
tains its  greatest  value  at  the 
point  of  inflexion,  and  the 
trace  allows  us  to  determine  this  maximum  value,  corre- 
sponding with  the  end  of  the  compression  of  the  shock  for 
each  experiment. 

We  have,  then,  given  the  mass  M  of  the  impinging 
body,  the  velocity  of  its  arrival  upon  the  body  shocked, 
due  to  the  height  of  fall,  the  mass  of  the  body  M'  impinged 
upon,  whose  initial  velocity  Y'  is  nothing,  and  by  obser- 
vation the  common  velocity  which  the  two  bodies  assume 
after  the  shock. 

It  is  easy  to  compare  in  each  case  the  results  of  exper- 
iments with  those  of  theory.  Some  of  these  comparisons 
are  presented  here. 


upon  the  transmission  of  motion  ~by  the  shock 
of  a  spherical  projectile  falling  upon  a  box  filled  with 
day,  or  containing  pieces  of  wood. 


Ij 

1 

*? 

1 

i  . 

• 

•3 

Velocity 

2|    . 

Weight  of 
Box  and  i 
load. 

Weight  of 
Sphere. 

•£* 

Height  of 
of  Sphere 

;ity  due  to 
fht  of  fall. 

imparted 
to  box. 

Hi 

l-s-s 

£ 

.    55 

P 

P 

P+P 

h 

li- 

2 

*t 

111 

IbB. 

Ibs. 

Ibs. 

ft. 

ft. 

ft. 

ft. 

sec. 

182.84 

(    1  O  OQ 

146.08 

1.81 

9.19 

0.88 

0.85 

.012 

•) 

184.74 

>•  Iz.^o 

147.97 

1.64 
0.65 

10.28 
6.50 

0.92 
1.08 

0.93 
1.08 

.020 
.019 

Experiments  made  with  clay,  whose 
resistance  to  penetration  of  pro- 

132.84 

26.64 

159.28 

0.98 

7.96 

1.40 

1.81 

.021 

[      jectiles  with  small  velocity  was 

1.81 

9.19 

1.52 

1.51 

.024 

6080  Ibs.  per  square  foot. 

182.84 

44.78 

177.57 

0.65 

6.50 

1.64 

1.61 

.020 

147.82 
147.82 

12.23 
44.73 

161.05 
192.54 

0.65 
0.65 

6.50 
6.50 

0.53 
1.51 

0.54 
1.44 

.063 
.072 

1  Experiments  made  with  clay,  whose 
V      resistance  to  penetration  was  34G 
)       Ibs.  to  sq.  ft. 

48.45 

26.64 

74.85 

0.65 

6.50 

2.27 

2.16 

.0075 

48.45 

26.64 

74.85 

0.98 

7.96 

2.79 

2.75 

.0074 

48.45 

44.73 

98.14 

0.38 

4.59 

2.20 

2.26 

.0060 

We  see  by  the  results  entered  in  the  above  table,  that 
the  velocities,  so  far  as  we  are  able  to  verify  them  with 


TRANSMISSION   OF  MOTION  BY   FORCES.  73 

such  means,  are  the  same  as  those  deduced  from  the  pre- 
ceding theoretic  considerations. 

68.  Shock  of  two  elastic  bodies.  —  If  we  suppose  that 
the  two  bodies  in  consideration  are  perfectly  elastic,  the 
effects  of  compression  will  be  at  first  the  same  as  in  the 
preceding  case,  and  at  the  end  of  this  time  the  body  M 
will  have  lost  a  velocity  Y—  U,  or  a  quantity  of  motion 
M(Y—  U)  and  the  body  M'  will  have  gained  a  quantity 
of  motion  M'(U—  Y'),  and  the  quantities  being  then  equal, 
we  have  for  the  common  velocity  at  the  end  of  the  com- 
pression 

V__MY+M/V/ 

M+M' 

But,  after  the  instant  of  greatest  compression,  the  elastic 
bodies  regain  their  primitive  form,  and  in  the  return  to  it 
develop,  if  the  elasticity  is  powerful,  efforts  equal  to  their 
resistance  or  compression,  and  consequently  destroy  or 
impart  quantities  of  motion  equal  to  those  which  they 
have  previously  destroyed  or  imparted.  It  follows  from 
this  that  in  the  unbending  of  the  molecular  springs  the 
body  M  will  further  lose  a  velocity  =Y—  U,  and  that  its 
final  velocity  will  be 

V_2(Y-U)=2U-Y, 

and  that  the  body  M'  will  receive  a  new  increase  of  ve- 
locity equal  to  U—  Y',  and  will  then  have  a  final  velocity 
equal  to 


If  the  body  were  at  rest  at  the  beginning,  in  supposing  it 
to  be  perfectly  elastic,  it  will  then  receive  a  velocity 


._ 

~M+M'* 

That  is  to  say,  twice  that  imparted  to  a  soft  body  in  the 
same  circumstances. 


TRANSMISSION   OF   MOTION   BY   FOKCES. 


69. — Observations  upon  the  preceding  results. — The 
foregoing  reasonings  relative  to  soft  or  elastic  bodies  pre- 
supposes the  existence  of  bodies  deprived  of  all  elasticity, 
and  of  others  endowed  with  perfect  elasticity.  Now, 
neither  of  these  hypotheses  is  exact,  and  according  to  the 
circumstances  in  which  they  are  placed,  a  body  may  act 
as  if  deprived  of  all  elasticity,  or  as  if  possessed  of  only  a 
partial  elasticity.  So  also  a  body  which,  in  certain  con- 
ditions, acts  as  if  it  were  perfectly  elastic,  will  only  appear 
as  if  but  partially  so  in  other  cases. 

I  will  cite  as  examples  the  results  of  some  experiments 
analogous  to  the  preceding,  and  which  were  effected  by 
placing  at  the  bottom  of  a  movable  box  a  plate  of  cast- 
iron,  upon  which  fell  a  spherical  body. 

Experiments  upon  the  transmission  of  motion  l>y  the 
shock  of  a  spherical  projectile  falling  upon  a  cast- 
iron  plate. 


Velocity 

Weight  of 

Velocity 

imparted  to  the 

Approximate 

the  box 

Weight  of 

Height  of 

due  to 

Box. 

duration 

and 

the 

Total 

the  fall  of  the 

this 

By 

By 

of  the 

its  load. 

Iron  ball. 

Weight. 

ball. 

height. 

Theory. 

Experi- 

transmission. 

P 

P 

P+P 

h 

2  U 

ment. 

Ibs. 

Ibi. 

Ibs. 

ft. 

ft. 

ft. 

ft. 

seconds. 

185 

13.23 

148.24 

1.31 

9.19 

1.64 

1.64 

0.0085 

136 

13.23 

148.24 

1.64 

10.28 

1.84 

1.87 

0.0081 

135 

13.23 

148.24 

1.9T 

11.26 

2.07 

2.05 

0.0080 

185 

26.44 

161.45 

1.31 

9.19 

3.01 

2.98 

0.0065 

185 

26.44 

161.45 

1.64 

10.28 

3.36 

3.44 

0.0075 

The  results  recorded  in  this  table  show  that  the  cast- 
iron  plate  shocked  has  acted  as  a  body  perfectly  elastic. 
But  it  is  proper  here  to  make  some  important  remarks. 

The  projectile  which,  (had  it  been  in  the  condition  of 
a  perfectly  elastic  body,)  as  well  as  the  parts  of  the  plate 
with  which  it  came  in  immediate  contact,  would  have 
risen  a  height  corresponding  to  the  velocity  2U— Y,  did 
not  by  any  means  attain  this  height.  This  proves  that 
the  intensity  of  the  shock  in  these  experiments  had  changed 


TRANSMISSION  OF  MOTION  BY  FOECES.  75 

in  a  great  measure  the  elasticity  of  the  molecular  springs 
of  the  parts  in  contact,  while  the  elasticity  of  flexure  or 
of  the  general  form  of  the  plate  had  not  been  altered. 
"We  see  by  this,  that  although  bodies  endowed  with  a  cer- 
tain elasticity  apparently  resume  their  primitive  form, 
there  is  nearly  in  every  case  a  notable  loss  of  work  pro- 
duced by  the  shock,  by  reason  of  the  more  or  less  com- 
plete alteration  of  its  elasticity.  We  shall  see  this  more 
explicitly  stated  hereafter  in  JSTo.  95. 

70.  Quantity  of  motion  imparted  by  a  constant  force. — 
When  the  force  is  constant  we  have  FT=MY,  whence 

MY 
Y=-=r.    This  expression  shows  that  the  effort  required 

to  impart  or  destroy  a  given  quantity  of  motion  MY  is  so 
much  the  greater  as  the  time  employed  is  less,  and  since 
the  reciprocal  action  of  bodies  is  more  rapid  as  the  spaces 
described,  their  compressions,  flexures,  and  penetrations 
are  less  for  the  same  quantity  of  motion  destroyed.  We 
have  here  explained  why  it  is  that  the  shock  of  hard 
bodies,  the  transmission  or  destruction  of  motion  by  bodies 
slightly  flexible,  compressible  or  extensible,  occasion  such 
great  efforts  and  such  ruptures  and  accidents,  and  how  it 
is,  on  the  other  hand,  by  the  interposition  of  soft  and  com- 
pressible bodies  that  the  intensity  of  efforts  and  their  con- 
sequences is  so  much  diminished. 

We  see  by  the  expression  F=-=-  that  a  finite  velocity  Y 

could  never  be  imparted  in  an  infinitely  small  time  (nul) 
to  a  mass  M  except  by  an  infinite  effort,  which  shows  the 
error  in  the  hypothesis  of  the  instantaneous  transmission 
of  motion  by  forces,  to  which  we  are  then  compelled  to 
give  a  special  name,  and  thus  suppose  a  special  nature  in 
calling  them  forces  of  percussion :  this  error  is  often  too 
explicitly  admitted  in  the  teachings  of  rational  mechanics. 
Nothing  like  an  instantaneous  operation  really  occurs  in 
nature  ;  quantities  of  motion  are  imparted  and  destroyed 


76  TRANSMISSION   OF  MOTION   BY  FOKCES. 

in  greater  or  less  periods  of  time,  sometimes,  indeed,  im- 
perceptible to  our  senses  and  means  of  observation,  but 
never  instantaneous.  The  idea  of  percussion  is  then 
erroneous  in  itself,  if  regarded  in  the  sense  just  indicated. 
Examples  will  enable  us  to  better  appreciate  this  mat- 
ter. In  case  we  require  the  quantity  of  motion  imparted 
to  a  ball  weighing  26.46lb%  and  upon  which  gunpowder 
has  impressed  a  velocity  of  1  640.4"-  in  1",  we  have 


FT=0.8222  x 
If  we  suppose  successively 

T=1.00",  0.50",  0.10",  0.01", 
we  have 

F=1348.7lb%  2697.4lb%  13487.lbs-,  134870lb8- 

The  velocity  being  communicated  in  less  than  Ti¥  of 
a  second,  gives  us  an  idea  of  the  enormous  efforts  de- 
veloped by  powder,  though  we  have  regarded  it  but  as  a 
mean  constant  effort,  and  consequently  far  inferior  to  the 
maximum  value  of  the  real  effort. 

When  horses  impress  upon  a  coach  weighing  9924  lbs- 
a  velocity  of  32808ft-  per  hour,  or 


the  quantity  of  motion  to  be  imparted  is 


WQ  have,  then, 

FT=2809.2. 


TRANSMISSION   OF  MOTION  BY   FOECES.  77 

If  we  suppose  that  each  one  of  the  five  horses  exerts 
in  any  time  a  mean  effort  of  220.5lb%  we  shall  have 


neglecting  the  resistance  of  the  ground,  and  the  friction 
of  the  wheel-boxes,  which  in  common  cases  would  require 


an  effort  of 
9924 


30 


=330.8lb%  or  of  66.2lb8-  per  horse. 


"We  see,  in  this  case,  that  to  impart  this  velocity  in 
2. 55",  each  horse  must  develop  a  mean  effort  of  about 
286.7lb%  which  is  more  than  four  times  the  mean  effort  to 
be  exerted  after  the  velocity  has  been  once  acquired. 

It  is  proper  to  observe  here,  that  the  breaking  of  traces, 
of  swing  bars,  wounds  upon  the  breasts,  and  straining  of 
hams,  arises  from  the  great  rapidity  of  the  destruction 
of  the  quantity  of  motion  impressed  by  the  horses  upon 
their  own  mass  by  the  resistance  and  reaction  of  the 
inertia  of  the  vehicle :  whence  the  necessity  of  starting 
with  slack  traces,  and  of  warning  and  urging  the  horses 
gently  with  the  voice. 

Similar  effects  are  produced  in  starting  and  stopping 
railroad  trains ;  and  in  seeking  the  means  of  promptly 
checking  these  enormous  masses,  we  must  bear  in  mind 
that  too  sudden  changes  of  velocity  are  dangerous  for  the 
passengers. 

Finally,  the  means  adopted  for  the  connection  of  ma- 
chines, or  for  a  rapid  transmission  of  motion  to  them, 
should  be  disposed  or  proportioned  agreeably  to  these 
ideas. 

Jugglers,  clowns,  herculean  fellows,  in  their  feats  of 
skill  or  strength,  are  led  by  observation  to  a  practice  con- 
forming to  that  above  indicated,  and  they  are  never  seen 
to  raise,  hurl,  or  arrest  very  heavy  weights,  or  make  their 


78  TRANSMISSION   OF  MOTION  BY  FORCES. 

jumps  suddenly,  but  always  gradually  increasing  the 
time  and  the  spaces  described,  so  as  to  dimmish  the  ef- 
forts. 

71.  Observations  upon  the  use  of  quantity  of  motion. — 
When  we  know  the  product  of  the  mass  of  a  body  and 
the  velocity  imparted  or  taken  from  it,  we  have  the  meas- 
ure of  effect  produced  by  the  force  during  the  period  of 
its  action ;  but  we  see  that  this  measure  cannot  be  taken 
as  a  term  of  comparison  except  for  analogous  cases,  where 
the  velocities  are  really  imparted  or  destroyed  by  the 
force,  and  it  does  not  follow  that  the  product  FT  of  the 
force,  by  its  period  of  action,  (equal,  when  there  is  a  change 
in  the  state  of  motion,  to  the  quantity  of  motion  imparted 
or  destroyed,)  should  always  serve  as  a  measure  of  the 
effect  of  forces,  as  is  sometimes  admitted  for  certain  in- 
struments and  for  certain  kinds  of  work.  Indeed,  it  is 
readily  seen  that  an  effort  may  continue  a  long  time  with- 
out producing  a  mechanical  effect.  Thus,  horses  pulling 
upon  a  mired  wagon  without  starting  it  develop  consider- 
able efforts,  which,  multiplied  by  the  period  of  their 
action,  would  give  an  enormous  product  without  any  use- 
ful effort  resulting,  any  mechanical  work,  and  nothing 
but  the  fatigue  and  exhaustion  of  the  motors. 

Take,  for  example,  the  draught  of  a  plough,  which  in 
strong  earth  requires  a  mean  total  force  of  794  pounds. 
We  suppose  the  furrow  to  be  393. 7ft  long,  the  horses  in 
one  take  100"  and  in  the  other  200"  to  plough  it.  We 
shall  have  for  the  first  case  FT=794lb8-xlOO"=79400, 
and  for  the  second  FT=794lbs-x  200" =158800,  and  yet 
in  both  cases  they  have  accomplished  the  same  work. 

An  instrument  giving  the  product  of  efforts,  by  the 
times  or  periods  of  duration,  would  by  no  means  lead  to 
an  exact  appreciation  of  the  mechanical  effects  produced. 
The  true  measure  of  these  effects  is,  as  we  have  said,  the 
product  of  the  effort  exerted  by  the  path  described  in  it8 
direction. 


TRANSMISSION    OF   MOTION   BY   FORCES.  79 

72.  Important  observation. — We  should  here  observe, 
that  it  is  only  in  the  case  of  a  constant  effort  acting  during 
a  time  T=lf/  that  we  can  take  the  product  MY  for  the 
measure  of  exerted  effort  F,  and  then  we  have 

F=MV=? V,  or  F  :  P  :  :  Y  :  g, 

y 

a  proposition  resulting  directly  from  the  general  principle 
enunciated  in  No.  61.  But  in  the  case  of  variable  efforts, 
the  same  mode  of  measurement  does  not  apply  for  finite 
times,  for  forces  varying  according  to  very  different  laws 
may  in  the  same  time  impart  equal  quantities  of  motion 
to  the  same  body  or  to  different  bodies.  The  formula 
F=MV  will  only  give  then  the  value  of  a  mean  con- 
stant effort  capable  of  imparting  in  the  same  time  the 
same  quantity  of  motion. 


OBSEKVATION  OF  THE  LAWS  OF  MOTION. 

73.  Determination  of  the  intensity  of  forces  ty  observ- 
ing the  laws  of  the  motions  they  produce. — The  formula 


shows  that  if  by  observation  of  the  laws  of  motion  we 

ft\ 

know  for  each  instant  the  value  of  the  ratio  -,  we  shall 

t 

then  have  that  of  the  corresponding  effort  F.  If,  for  exam- 
ple, we  know,  by  experiment,  that  the  motion  is  uniformly 
accelerated,  we  have 


whence 

Vi.   v  «,=ss 

consequently 


If,  for  instance,  a  wagon  weighing  2205  pounds  runs 
with  a  uniformly  accelerated  motion  a  distance  of  32.8ft-  in 
2",  we  have 

-        2205       2x32.8 


OBSERVATION   OF   THE  LAWS   OF  MOTION-  81 

. 

for  the  value  of  the  force  capable  of  imparting  this  accel- 
erated motion,  deduction  being  made  for  friction. 

74.  Means  employed  for  determining  the  laws  of  motion 
of  bodies. — According  to  the  nature  of  the  case,  we  make 
use  of  different  contrivances  or  instruments  for  observing 
the  laws  of  motion  of  bodies.     For  a  slow  motion,  we 
employ  watches,  pendulums,  and  note   the  time    corre- 
sponding with  the  given  spaces.     But  for  rapid  motion 
these  methods  do  not  afford  the  requisite  precision. 

75.  Colonel   Beaufoy's  apparatus. — (Fig.   30.) — This 


v  U  ;:,„„«  El  m 

FIG.  30. 

experimenter,  in  his  researches  upon  the  resistance  of 
water,  was  provided  with  a  pendulum  which  traced  at 
each  oscillation  a  mark  upon  a  rule,  whose  motion  was  in 
a  known  ratio  with  that  to  be  observed,  and  as  in  his  ex- 
periments the  motion  soon  became  uniform,  the  velocity 
of  the  motion  was  easily  obtained. 

76.  Eytelweirfs  apparatus. — This  learned  engineer 
seems  to  have  been  the  first  to  entertain  the  idea  of  com- 
bining a  known  uniform  motion  with  that  to  be  deter- 
mined, so  as  to  obtain  a  trace  of  simultaneous  motions, 
from  which  might  be  deduced  the  condition  of  the  un- 
known motion. 

For  his  experiments  upon  the  hydraulic  ram,  he  used 
an  endless  band  of  paper,  (Figs.  31  and  32,)  rolled  upon 
two  cylinders,  to  which  a  regular  motion  was  imparted 


OBSERVATION   OF    THE   LAWS   OF   MOTION. 


by  the  hand.     The  lengths  of  the  paper  passed  were  then 
nearly  proportional  to  the  time.     A  style  fixed  to  the 


J_L 


II     , 

r^ 


=L  L 


Fig.  31. 


stem  of  the  valves  traced  upon  this  band  a  curve  whose 
ordinates  were  the  spaces  described. 

M.  Eytelwein  could,  by  means  of  this  imperfect  con- 
trivance, nearly  determine  the  intervals  of  time  between 
the  opening  and  shutting  of  the  valves. 

But  we  know  that  the  motion  imparted  by  the  hand 
could  not  have  been  uniform,  and  that  this  disposition 
could  not  give  very  exact  results. 

77.  New  apparatus. — For  experiments  made  at  Metz, 
upon  friction,  and  for  other  researches,  at  M.  Poncelet's 
suggestion,  I  made  use  of  a  combination  of  a  known  mo- 
tion with  one  whose  law  was  to  be  determined. 

I  have  since  modified  the  arrangement  of  this  machine, 
and  that  deposited  in  the  repository  of  arts  and  machines 
was  made  in  the  following  manner,  (Figs.  33  and  34 :) 
a  plate  1.05ft-  in  diameter,  perfectly  plane,  receives  a 
uniform  motion  by  means  of  a  weight  hung  to  a  first  axle. 
The  motion  of  this  axle  is  transmitted  to  a  second  axle  by 
a  wheel  and  pinion,  whose  number  of  teeth  are  to  each 
other  as  6  :  1.  A  wheel  mounted  upon  the  second  axle 
drives  a  second  pinion  fixed  upon  the  axle  of  the  plate. 
This  wheel  and  its  pinion  have  also  their  number  of  teeth 


OBSERVATION    OF    THE    LAWS    OF   MOTION. 


83 


in  the  ratio  of  6  :  1,  so  that  the  plate  makes  36  turns  for 
one  of  the  first  axle. 


Elevation. 


FIG.  38. 


Fio.  34. 


Upon  the  axle  of  the  plate  is  a  fly-wheel  with  4  wings, 
0.33ft-  at  the  sides,  which  serves  to  regulate  the  motion  by 
the  resistance  of  the  air,  which  is,  as  we  know,  nearly 
proportional  to  the  square  of  the  velocity. 

It  follows  from  this  arrangement,  that  in  a  short  time 
the  motion  of  the  plate  becomes  uniform,  its  centre  of 
gravity,  as  well  as  that  of  the  other  pieces,  being  upon 
the  axis  of  rotation. 

This  uniformity  may  be  readily  ascertained  by  taking 
the  number  of  turns  of  the  second  axle,  as  indicated  by 
the  pointer. 

Just  opposite  the  plate,  and  parallel  to  its  surface,  is  a 
pulley,  whose  motion  is  in  a  known  ratio  with  that  to  be 
observed.  The  axle  of  this  pulley  bears  a  small  arm, 
upon  which  is  placed  a  style,  formed  usually  of  a  brush 
filled  with  India  ink. 

By  simple  means  we  test  the  parallelism  of  the  circle 
described  by  the  point  of  the  style  with  the  surface  of  the 


OBSERVATION   OF    THE   LAWS   OF  MOTION. 


plate,  upon  which  is  fastened  a  sheet  of  paper.  The  style 
may,  before  the  experiment,  be  kept  at  a  short  distance 
from  the  paper,  and  be  brought  in  contact  with  it  the  in- 
stant the  motion  begins. 

We  easily  conceive,  after  this  description,*  that  the 
plate  turning  with  a  uniform  motion,  and  the  style  with 
an  unknown,  there  follows  from  these  simultaneous  mo- 
tions a  trace  left  upon  the  paper,  which,  depending  upon 
the  synchronous  motion,  should  give  in  the  tabulations 
the  relation  of  angles  described  by  the  pulley,  or  of  the 
spaces  described  by  the  observed  body,  with  the  angles 
described  by  the  plate,  or  their  corresponding  times. 

This  readily  appears 
in  observing  that,  if 
the  plate  were  at  rest, 
the  style  would  de- 
scribe upon  its  surface 
a  circle  with  a  radius 
equal  to  its  distance 
from  the  axis  A  of  the 
pulley.  While,  on  the 
other  hand,  if  Jhe  style 
were  at  rest,  and  the 
plate  in  motion,  the 
latter  would  have  for 
the  trace  of  its  contact 
with  the  brush,  a  circle  whose  radius  is  that  of  the  plate, 
and  its  radius,  the  distance  of  the  style  from  its  centre. 
This  granted,  let  o  be  the  origin  of  the  curve  traced 
during  the  experiment.  Through  this  point  describe  a 
circle  with  a  radius  Ao,  equal  to  the  distance  of  the  style 
from  the  axis  of  the  pulley,  with  its  centre  at  a  known 
distance  Ac  from  that  of  the  plate,  and  divide  this  circle 
into  ten  equal  parts  at  the  points  0,  1,  2,  3  ...  9. 


FIG.  86. 


*  For  further  details,  see  the  description  "  des  Appareils  chronometriquesj 
inserted  in  the  journal  of  the  scientific  convention  held  at  Metz  in  1837. 


OBSERVATION    OF    THE   LAWS   OF   MOTION.  85 

Through  each  of  these  points  we  draw  circumferences 
of  circles,  with  their  centres  at  c,  radii  CO,  01,  02,  &c. 
These  circles  will  cut  the  curve  in  the  points  1',  2',  3',  &c. 

]STow,  it  is  evident  that  the  point  V  results  from  the 
simultaneous  motions  of  the  style  from  0  to  1,  and  of  the 
plate  describing  the  angle  101'  ;  consequently,  the  arc  01 
gives  the  angle  described  by  the  pulley,  or  the  space  de- 
scribed by  the  body,  and  the  angle  101'  furnishes  the 
measure  of  the  corresponding  time.  We  may  then  suc- 
cessively observe  these  spaces,  and  from  them  make  a 
table  representing  the  law  of  motion. 

Taking,  then,  the  spaces  described  for  the  abscissae, 
and  the  times  for  the  ordinates,  we  shall  have  a  curve 
with  rectangular  co-ordinates,  the  nature  of  which  must  be 
studied  to  derive  the  law  of  the  observed  motion. 

If,  for  example,  the  abscissae  or  the  spaces  described 
are  proportional  to  the  squares  of  the  times,  or  the  ordi- 
nates, the  curve  will  be  a  parabola,  and  the  motion  will 
be  uniformly  accelerated.  If  the  curve,  either  at  its  ori- 
gin, or  after  a  certain  time,  should  change  into  a  straight 
line,  from  that  instant  the  motion  will  be  uniform. 

When  the  motion  is  very  rapid,  for  which  styles 
charged  with  ink  are  not  suitable,  we  may  use  metallic 
styles  to  trace  upon  soft  materials,  such  as  wax  mixed 
with  tallow.  Thus  we  may  easily  determine  the  law  of 
motion  in  the  cock  of  a  musket,  though  this  motion  is 
made  in  nearly  T^F  of  a  second. 

We  begin  by  tracing  upon  the  plate  at  rest  the  arc  of 
the  circle  described  by  the  style  fixed  in  the  head  of  the 
cock,  and  formed  of  a  light  steel  pointer,  which  serves  to 
determine  upon  the  plate  a  circle  of  a  radius  CA,  upon 
which  is  projected  at  A  the  centre  of  the  tumbler.  This 
done,  we  set  the  plate  in  motion,  and  when  by  direct  ob- 
servation we  have  obtained  the  uniform  velocity  of  its 
motion,  we  let  go  the  pointer  and  the  cock,  and  obtain  a 
curve  0,  1,  2,  3,  4.  The  origin,  o  of  this  curve  can  be 
nearly  determined  at  first  sight  by  examining  its  point  of 


86 


OBSERVATION   OF    THE   LAWS   OF  MOTION. 


tangency  with  the  arc  of  the  circle  traced  by  the  style 
before  the  starting  of  the  cock.  We  trace  the  circle  de- 
scribed by  the  point  and  passing  through  the  origin  o. 
This  arc  terminates  at  the  circumference  which  the  style 
had  traced  when  the  cock  was  arrested. 

"We  divide  it  into  any  number  of  parts,  or  rather  start- 
ing from  the  point  0,  we  take  arcs  01,  02,  equal  to  a  given 
number  of  degrees,  and  corresponding  consequently  to 
known  angles  described  by  the  cock. 


FIG.  80. 

Then  from  the  point  C  as  a  centre,  and  with  radii  Cl, 
C2,  C3,  etc.,  we  describe  arcs  of  circles,  meeting  the  curve 
in  1',  2',  3',  etc.  Finally,  the  angles  101',  2C2',  3C3',  give 
the  corresponding  times. .  We  may  then  compare  the  an- 
gles described  by  the  cock  with  the  time  employed.  We 
find  thus,  for  example,  for  the  infantry  percussion  mus- 
kets, modelled  in  1840,  the  following  results  : 


Arcs  described 
by  the  centre 
of  the  coun- 
tersink   

Corresponding 

ft. 
0.01T38 

sec. 
0  00406 

ft. 
0.0347 

sec. 
0  00604 

ft. 
0.0525 

sec. 

000754 

ft. 
0.0698 

sec. 
*  OOS86 

ft. 
0.0741 

sec. 
000936 

ft. 
0.1046 

sec. 
0  01050 

ft. 
0.1230 

sec. 

0  01126 

ft. 
0.1397 

sec. 

0  01190 

ft. 
0.1571 

sec. 
0  01372 

ft. 
0.1706 

sec. 
0  01318 

Ratios  of  the 
squares  of  the 
time  to  the 
arcs  described. 

0.00094 

0.00105 

0.00108 

0.00112 

0.00118 

0.00105 

0.00103 

0.00101 

0.00120 

0.00103 

**  Mean. 


.0.00107 


*  In  ray  edition  it  is  written  0.08864 — probably  a  misprint. — TRANSLATOR. 
**  In  the  last  column  of  ratios,  Morin  has  apparently  made  some  error ; 
the  mean,  as  he  gives  it,  for  metres,  is  .00341.     It  should  be  .00351. 


OBSERVATION   OF    THE   LAWS   OF  MOTION.  87 

^Repeating  the  experiment  thrice,  a  mean  ratio  was 
found 

rp  r> 

|  =.0010759=^-, 

and  the  total  arc  being  0.1706ft-,  we  find  T=0.01355", 

and  so 


__  _ 
.0010759' 

and  finally 


for  the  velocity  of  the  style.  This  style  was  0.2001ft-  from 
the  axis  of  the  tumbler,  while  the  centre  of  the  counter- 
sink was  but  0.1975ft-,  consequently  the  velocity  of  the 
cock  at  this  centre  was 


r  x25.1Sft-  in  1  second. 
0.2001 

We  see  by  this  example  that  we  can  determine  a  great 
many  points  of  the  curve  which  represent  the  laws  of 
motion,  and  as  the  ratio  of  the  spaces  described  to  the 
squares  of  the  time  is  constant,  it  follows  that  this  curve 
must  be  a  parabola,  or  that  the  motion  of  the  cock  is  uni- 
formly accelerated.  Thus  the  force  which  produced  it  is 
constant,  and  the  form  of  the  curve  of  the  tumbler,  as 
well  as  the  resistance  of  the  spring  to  flexure,  are  so  com- 
bined that  the  effort  of  the  thumb  to  cock  the  gun  is  con- 
stant. This  shows  how  the  art  of  the  mechanic  may 
sometimes  prove  the  solution  of  quite  difficult  mechanical 
problems. 

We  may  extend  the  use  of  these  instruments  to  the 
observation  of  still  more  rapid  motions  ;  for,  in  the  ex 
periments  upon  the  cock  of  the  gun,  the  plate  only  made 
6  turns  in  1",  and  we  could  easily  obtain  10.  Then  the 
circumference  of  this  plate  being  3.281"-,  would  run 


88  OBSERVATION    OF    THE   LAWS    OF   MOTION. 

32.81ft-  in  1",  and  as  by  the  instrument  for  observing 
curves  we  can  appreciate  0.00065"-,  we  may  then  obtain 

the  time  to  nearly  ^7^^:  of  a  second. 


"We  might  go  still  further  by  increasing  the  dimensions 
of  the  plate. 

If  we  should  furthermore  combine  the  movement  of 
the  plate  with  electricity,  we  might  perhaps  determine 
the  duration  of  certain  phenomena  so  rapid  in  their  dura- 
tion as  thus  far  to  have  eluded  all  our  attempts  at  meas- 
urement. 

Among  these,  for  example,  would  be  the  law  of  mo- 
tion of  projectiles  in  the  air,  for  the  solution  of  which 
attempts  have  already  been  made,  but  with  small  success, 
since  instead  of  an  apparatus  with  continuous  motion,  we 
have  used  chronometric  instruments,  with  an  oscillating 
or  intermittent  motion. 

78.  Zinc  plates.  —  The  sheet  of  paper  which  usually 
receives  the  traces  of  the  style  is  wet  and  pasted  at  its 
edges  upon  a  zinc  plate,  which  is  fastened  to  the  plate. 
This  avoids  the  inconvenience  of  unequal  shrinkage  of 
the  paper,  and  greatly  facilitates  the  tabulations. 

79.  —  Contrivance  for  tabulating  the  curves.  —  The  sheet 
of  zinc,  when  taken  from  the  plate,  being  exactly  centred, 
is  placed  upon  an  instrument  for  making  abstracts  of  the 
curve.     The  circumference  of  this  instrument  is  divided 
into  a  thousand  parts.     An  alidade  movable  around  its 
axis,  bears  a  disc  provided  with  ten  pointers,  whose  ex- 
tremities divide  into  ten  equal  parts  the  circumference 
described  by  the  style  upon  the  plate,  while  at  rest.     We 
loosen  the  thumb-screw  which  fastens  this  disc,  and  then 
turn  it  round  so  that  one  of  these  points  shall  correspond 
with  the  origin  of  the  curve.     This  done,  we  fasten  the 
disc  firmly  to  the  alidade  with  the  thumb-screw.     Since, 
then,  it  is  evident  that  each  of  these  ten  points,  in  the 


OBSERVATION   OF    THE   LAWS    OF   MOTION.  89 

movement  of  the  alidade,  will  describe  the  auxiliary  circle 
11',  22',  33',  99',  etc.,  by  turning  the  alidade  so  that  all 
the  points  may  successively  meet  the  curve,  and  reading 
the  angles  described  by  the  alidade  corresponding  to  each 
point,  we  shall  have  the  times  which  are  proportional  to 
these  angles,  and  the  angles  described  by  the  pulley  of 
the  style,  by  the  number  in  order  of  the  points. 

This  instrument,  for  which  we  are  indebted  to  M. 
Didion,  Captain  of  Artillery,  combines  great  precision 
and  utility  in  all  operations  of  this  kind. 

These  contrivances  just  described  and  constructed  by 
me,  are  the  first  of  this  kind,  and  are  conveniently  used 
in  various  experiments,  but  the  results  require  an  elabo- 
rate abstract,  and  do  not  address  the  eye  quickly  enough 
for  the  purposes  of  instruction.  It  has  for  some  time 
seemed  desirable  to  apply  the  principle  of  this  construc- 
tion in  a  more  simple  form.  It  was  thus  that  I  was  led 
to  the  construction  of  the  following  apparatus : 

80.  Description  of  a  chronometric  apparatus  with  cyl- 
inder and  style,  for  observing  the  laws  of  motion. — The 
principal  piece  of  this  apparatus,  in  the  model  adopted 
for  the  Lyceums  by  the  minister  of  public  instruction,  is  a 
vertical  cylinder  AB,  (fig.  37),  6.89ft-  high,  covered  with 
common  white  paper  slightly  wet,  and  pasted  at  its  edges. 
This  cylinder  is  0.41ft-  in  diameter,  answering  to  a  circum- 
ference of  1.2S6ft>.  It  rests  upon  a  pivot,  and  is  put  in 
motion  by  means  of  a  contrivance  similar  to  a  kitchen- 
jack. 

A  weight  is  suspended  to  a  cord  rolled  round  the 
surface  of  a  drum  C.  At  one  end  of  this  drum  is  a  wheel 
D  with  teeth,  inclined  45°,  which  drives  an  endless  screw, 
whose  vertical  axis  carries  at  its  upper  end  a  fly-wheel 
with  wings,  which  serves  to  regulate  the  motion  by  means 
of  the  resistance  of  the  air.  We  may  incline  these  wings 
to  increase  or  diminish  the  regulating  action  of  the  air, 


90 


OBSERVATION   OF    THE  LAWS   OF  MOTION. 


and  as  we  may  also  vary  the  motive  weight,  we  may  ob- 
tain a  uniform  motion  of  the  cylinder  at  a  velocity  of  one 
turn  per  second,  or  a  more 
rapid  motion  if  desired. 
The  motion  being  regulat- 
ed gradually,  it  is  well  not 
to  make  the  projected  ex- 
periment before  the  motive 
weight  has  run  through 
from  |  to  |  of  its  fall; 
there  remains  more  than 
time  enough  for  ordinary 
cases. 

There  is  at  the  Con- 
servatory of  Arts  and  Ma- 
chines a  much  larger  and 
more  complete  model  of 
this  instrument,  whose  cyl- 
inder is  10.17ft-  high  and 
3.2Sft-  in  circumference. 

If  the  circumference 
of  the  upper  and  lower 
base  of  the  cylinder  is  di- 
vided into  100  equal  parts, 
each  of  these  points  wilt 
correspond  to  -fi¥  of  a 
revolution  or  of  a  second, 
and  as  each  of  these  may 
be  divided  into  10  parts  of 
0.032f%  we  see  that  each 
of  them  aifords  the  means 
of  measuring  the  time  with 
the  precision  of  Ti-0-  or 
T  oV  o  of  a  second,  and  even 
less.  Here,  then,  we  have  FIG.  ST. 

a  very  delicate  chronometer.    The  division  of  which  we 


OBSERVATION   OF    THE   LAWS   OF  MOTION.  91 

have  spoken  is  easily  effected,  by  means  of  a  wooden  rule 
placed  upon  a  post  parallel  to  the  edge  of  the  cylinder 
and  near  its  surface.  The  base  of  the  cylinder  has  a  cir- 
cle, with  100  equidistant  ratchet  teeth,  in  each  of  which 
falls  a  catch  to  fasten  the  cylinder  while  the  generatrix  is 
traced  with  the  rule.  The  rule  is  also  divided  at  spaces 
of  0.16ft-  into  notches,  in  which  is  placed  a  crayon,  which 
is  held  fast  while  the  cylinder  is  turned,  so  as  to  trace 
parallels  to  the  base,  whose  developments  give  the  ordi- 
nates  of  the  curve  of  motion. 

Now,  suppose  a  body  M,  hung  near  the  summit  of  the 
cylinder  upon  a  bent  lever  ab,  is  left  to  fall  to  the  earth, 
guided  vertically,  by  means  of  two  metallic  threads,  well 
stretched  by  the  screws  v  v'  parallel  to  the  edge  of  the 
cylinder,  and  that  the  body  carries  a  style,  formed  of  a 
brush  soaked  in  ink,  or  rather  of  a  crayon  pencil  pressed 
against  the  surface  of  the  cylinder  by  a  spring,  we  see 
that  if  during  the  descent  of  the  body  the  cylinder  is  at 
rest,  the  style  will  leave  a  trace  of  the  generatrix  of  the 
cylinder,  or  of  a  straight  line.  But  as  the  cylinder  moves 
at  the  same  time  the  weight  falls,  the  style  traces  upon  the 
sheet  of  paper  a  curved  line  depending  upon  the  two 
simultaneous  motions. 


81.  Discussion  of  the  results  furnished  ~by  this  appa- 
ratus.— When  the  curve  has  been  obtained,  and  the  gen- 
eratrix of  the  cylinder  has  been  traced,  corresponding  to 
the  origin,  it  is  easy  to  recognize  its  nature,  and  to  prove 
that  the  motion  of  the  style  which  traced  it,  and  conse- 
quently that  of  the  body  which  bore  the  style,  was  uni- 
formly accelerated. 

Indeed,  if  we  cut  the  paper  and  take  it  from  the  cyl- 
inder, the  generatrix  drawn  through  the  origin  of  the 
curve  may  be  taken  for  the  axis  of  the  abscissae,  and  the 
lengths  upon  this  line,  measured  from  the  origin,  will  be 
of  the  same  magnitude  as  the  spaces  described  by  the 


92  OBSEKVATION   OF    THE   LAWS    OF   MOTION. 

body  in  its  fall.  The  ordinates  of  this  curve  will  be  the 
developments  of  so  many  arcs  of  the  circle  of  the  cylin- 
drical surface,  and  each  millimetre  =0.003ft-  of  these  or- 
dinates will  represent  a  given  fraction  of  the  second. 

We  shall  then  have  obtained  with  this  contrivance  a 
curve  whose  abscissae  are  the  spaces  described  by  the 
body,  and  its  ordinates  are  the  corresponding  times. 

Now,  comparing  directly  the  spaces  described  with 
the  times  measured  upon  the  curve,  we  see  at  once  that 
the  first  arc  is  in  a  constant  ratio  with  the  squares  of  the 
seconds,  which  shows  that  the  motion  of  descent  of  the 
body  was  uniformly  accelerated. 

By  a  simple  graphic  construction,  which  consists  in 
drawing  at  sight  with  a  ruler  a  series  of  tangents  to  the 
curve,  to  determine  the  point  where  they  cut  the  axis  OY 
of  ordinates,  and  in  raising  at  these  points  perpendiculars 
to  each  of  these  tangents,  we  find  that  all  these  perpen- 
diculars intersect  in  the  same  point,  a  property  of  the 
parabola :  a  curve  whose  abscissae  are  in  a  constant  ratio 
with  the  square  of  its  ordinates. 

The  point  thus  determined  is  the  focus  of  the  parabola, 
and  furnishes  the  true  position  of  the  axis  of  the  abscissae 
and  of  the  origin  of  the  curve,  positions  thus  far  supposed  to 
be  determined  by  the  eye,  and  naturally  attended  with 
some  uncertainty. 

82.  Determination  of  the  velocity. — Since  we  have 
determined,  directly,  and  by  the  instrument,  the  curve 
representing  the  law  of  motion  uniformly  accelerated, 
which  we  have  recognized  as  a  parabola,  we  shall  have 
the  velocities  of  motion  changing  at  each  instant,  by  the 
inclination  of  the  tangents  of  this  curve  with  its  ordinates. 
Now,  this  inclination  in  the  parabola  is  equal  to  double 

2S 
the  abscissae  S  divided  by  the  time  T,  or  to  -^  ;   we   have 

T7    2S      ,          Y     2S 
then  Y^-^,  whence  —  =  — ,  and  as,  by  comparison  of 


OBSERVATION   OF    THE   LAWS   OF   MOTION.  93 

the  abscissae  S  with  the  squares  of  the'ordinates  T,  we  have 

S  Y 

found  that  ^  is  constant,  we  see  also  that  the  ratio  „-  is 

constant,  which  shows  that,  agreeably  to  the  definition  of 
uniformly  variable  motion,  the  velocities  here  are  propor- 
tional to  the  times. 

If  we  seek  the  velocity  Yx  ac- 
quired at  the  end  of  the  first  second 
of  the  fall,  we  find  ¥,=28^ 

Sx  being  the  space  described  after 
the  first  second,  or  the  abscissae,  cor- 
responding to  the  ordinate  T=l". 

We  see,  then,  that  this  new  appa- 
ratus enables  us  to  determine  directly 
by  observation  all  the  circumstances  Fia  8g 

of  the  descent  of  bodies  falling  freely  . 
to  the  surface  of  the  earth,  to  prove  that  this  motion  is 
uniformly  accelerated,  and  to  obtain  also,  with  exactness, 
the  value  of  the  velocity  acquired  by  bodies  in  the  first 
second  of  their  fall,  which  at  Paris  is  equal  nearly  to 
32.1817ft-* 

83.  Experimental  demonstration  of  the  principle  of 
the  proportionality  of  forces  to  the  velocities. — This  prin- 
ciple which  we  have  admitted,  in  Art.  61,  as  the  resultant 
of  all.  the  observed  phenomena,  and  of  which,  till  the 
present,  no  direct  demonstration  has  been  given,  may  be 
easily  verified,  by  means  of  an  apparatus  made  jointly  by 
M.  Tresca  and  myself,  for  the  course  of  Mechanics  of  the 
Institute.  Imagine  a  movable  weight  P,  subject  to  the 
action  of  gravity,  to  be  connected  by  a  thread  with  another 
movable  weight  p,  free  to  move  upon  a  horizontal  plate, 
placed  firmly  near  the  apparatus.  The  action  of  gravity 
upon  the  second  body  will  be  destroyed  by  the  plate,  and 


*  The  models  of  this  cnronometric  apparatus,  made  for  the  Lyceums,  came 
from  the  workshops  of  M.  Glair,  machinist,  rue  du  Cherche-Midi,  93. 


94:  OBSERVATION    OF    THE   LAWS    OF   MOTION. 

when  the  body  P  descends,  it  is  solely  by  the  action  of 
gravity  upon  this  body  that  the  two  bodies  are  put  in 
motion,  and  so  from  the  force  P  alone,  that  motion  is  ini- 


parted  to  the  mass  —  -  .     In  these   conditions,  the  fall 

g 

will  take  place  with  a  uniformly  accelerated  motion,  but 
evidently  less  rapid  than  when  the  body  P  is  entirely 
free.  The  curve  traced  by  the  style  is  a  parabola,  more 
open  than  in  the  case  of  a  free  fall,  and  serves  to  demon- 
strate, as  we  have  before  observed,  the  velocity  Y  of  a 
system,  at  the  end  of  any  time,  a  second,  for  example. 
The  continuous  action  of  a  force  P  imparts,  then,  in  a 

second,  to  the  mass  ---  |r  a  velocity  Y  measured  experi- 

mentally. 

P'-fl/ 

We  may  also  observe  the  velocity  V  of  a  mass  --  =- 

j/ 

in  similar  circumstances,  by  substituting  a  weight  P'  for 
the  weight  P,  and  for  the  body  drawn  a  weighty/  instead 
of  p. 

Now,   in  this    twofold    substitution   we    may  make 
P'+2/,  in  which  case  the  mass  put  in  motion  in 


P+z?  P'-h0' 

both  cases  is  identical  :  --  —  being  equal  to  —  S-  ;   we 

t/  </ 

thus  realize  the  circumstance,  that  two  motor  weights, 
which  are  true  forces,  may  be  acting  the  same  time  upon 

the  same  mass   —  -  ;  and  observing  the  acquired  veloci- 

y 

ties  Y  and  Y7  in  the  two  cases,  all  that  is  needed  to  estab- 
lish the  principle  is  that  the  figures  give  the  proportion 

Y  :  Y'  :  :  P  :  F, 

which  is  an  interpretation  of  the  principle  of  the  propor- 
tionality offerees  and  velocities  already  enunciated. 


OBSERVATION    OF    THE   LAWS    OF   MOTION.  95 

These  experiments  require  care,  since  account  must  be 
rendered  of  frictions  and  of  the  passive  resistances,  in  dif- 
ferent parts  of  the  system. 

In  a  series  of  experiments,  the  mean  velocity  of  the 
cylinder  at  its  circumference  was  observed  while  the  style 
was  pressing  against  it  ;  it  was  found  to  be  1.5634"-  per 
second,  and  therefore  it  is  for  an  abscissa  of  this  length, 
that  the  velocity  should  be  shown  upon  the  curve  traced 
by  the  style. 

The  total  weight  was  14.393lb%  including  a  fraction  of 
0.498lb%  determined  by  previous  experiments  as  the  equiv- 
alent of  the  resistance  of  friction. 

The  motor  weights  were  successively        p  =0.540lbs- 

p'  =1.080 


and  the  corresponding  velocities  observed      Y  =1.161" 

V  =2.203 
V"=3.090 
The  ratio  of  the  motor  weights  being  : 

y_1.080  P"_ 

-~  - 


The  corresponding  ratios  between  the  observed  veloci- 
ties give 

V'2.203  V"     3.090 


The  close  approximation  of  these  ratios  shows,  in  the 
limits  of  experiment,  the  exactness  of  the  law,  which 
might  be  demonstrated  in  the  same  way  for  more  ex- 
tended limits. 


96  OBSERVATION   OF    THE   LAWS   OF   MOTION. 

The  velocities  calculated  a  priori,  according  to  the 
value  of  <7=32.18  would  be  respectively 

1.164:ft-        2.216ft-        3.117ft- 
which  differ  slightly  from  those  of  observation. 


PKISTCIPLE  OF  VIS  VIYA. 

84.  Measure  of  mechanical  work  developed  by  motive 
forces  or  inertia,  in  variable  motion.  —  We  have  already 
seen  that  the  motive  force  and  the  reaction  developed  by 
the  inertia  of  a  mass  M  impressed  with  a  parallel  arid 
variable  motion,  have  for  a  common  measure  the  ex- 
pression 


Consequently,  calling  s  the  elementary  space  described 
in  the  element  of  time  tf,  we  shall  have  for  the  elementary 
work  of  the  force  F 


We  would  remark  that  when  the  motion  is  accelerated, 
the  space  passed  over  by  the  point  of  application  of  the 
force  of  inertia,  then  acting  as  a  resistance,  is  described 
in  an  opposite  direction  to  the  force,  and  develops  a  resist- 
ance equal  to  the  work  of  the  applied  force  F.  On  the 
other  hand,  the  work  of  inertia  becomes  a  motive  force  if 
the  motion  is  retarded,  and  is  equal  to  that  of  the  force  F 
producing  the  diminution. 

Let  us  remember  that  in  variable  motion,  the  velocity 

o 

V  at  any  instant  is  equal  to  -,  so  that  the  above  expres 

t 

sion  becomes  F5=M.Y.  v. 

The  total  work  developed  by  the  motive  force  in  im- 
7 


98  vis  VIVA. 

parting  to  all  the  elements  of  the  body  P  or  of  the  mass 
p 
-  a  certain  common  velocity  V,  starting  from  repose,  is 

c/ 

then  the  sum  of  all  the  similar  elementary  quantities  of 
work.  Now,  if  we  place  the  velocities  upon  a  line  of  ab- 
scissae, and  raise  at  each  point  perpendiculars  equal  to  the 
abscissae  or  velocities,  it  is  clear  that,  for  an  elementary 
increase  v=ef  of  velocity,  the  pro- 
duct Vv  will  be  represented  by  the 
area  of  the  small  trapezium  ee'ff, 
and  that  the  sum  of  all  the  like 
products,  from  the  point  where 
V=0  to  V= AB=BB'  will  be  rep- 
resented by 

x  BB'^Y2,  so  that  the  total  work 

2i 

developed  by  the  motive  force,  or  the  work  developed  by 
the  force  of  inertia  will  be,  calling  it  "W  : 

W=Jbc  •  Vs.  =4  --VV 

2  2  g 

85.  Vis  Viva. — This  product  of  the  mass  by  the  square 
of  the  velocity  has  received  from  geometricians  the  con- 
ventional name  of  "  Yis  Viva." 

It  follows,  then,  from  the  preceding,  that  the  work  de- 
veloped by  a  force  which  imparts  to  or  takes  from  all  the 

p 
elements  of  a  body  with  a  mass  M=- ,  a  common  velocity 

V,  is  equal  to  one  half  of  the  vis  viva  corresponding  with 
this  velocity. 

If  the  body  is  impressed  with  a  certain  common  veloci- 
ty V,  or  with  a  vis  viva  MV/2  at  the  moment  when  the 
force  commences  its  change  of  motion,  it  is  evident  that 
the  force  can  only  have  imparted,  when  the  velocity  has 
become  Y,  or  its  vis  viva  MY2,  but  the  difference,  or  the 


vis  VIVA.  99 

excess  of  the  vis  viva,  which  it  finally  possessed  over 
that  which  it  had  at  the  commencement  of  its  action,  to 
wit,  MV2— MV'2  if  accelerated,  orMV/2-HV2  if  retarded, 
and  that  the  corresponding  work,  represented  by  the  dif- 
ference of  the  triangles  ABB'  and  ACC'  will  be  equal  to 


or 


V 

as  the  motion  is  accelerated  or  retarded. 

Thus,  in  general,  the  work  of  a  force  which  accelerates 
or  retards  the  motion  of  a  body  moving  in  its  own  direc- 
tion^ is  equal  to  one-half  the  vis  viva  which  it  has  im- 
parted to  or  taken  from  the  body. 

This  principle  has  received  the  name  of  the  vis  viva, 
and  its  generality  serves  as  a  base  for  all  applied  me- 
chanics. 

86.  Effects  of  the  gas  of  powder  in  fire-arms  and  ord- 
nance.— The  considerations  in  No.  65  and  the  following, 
relative  to  the  communication  of  the  quantity  of  motion, 
and  the  principle  of  vis  viva  apply  directly  to  the  effects 
of  explosive  gases  in  fire-arms,  with  an  approximation 
which  enables  us  to  deduce  useful  consequences. 

Indeed,  if  we  consider  what  occurs  in  the  short  inter- 
val of  the  flight  of  the  projectile,  and  suppose  the  charge 
so  small  as  that  the  inertia  of  its  gas  may  be  disregarded, 
we  may  admit  that  the  efforts  exerted  by  the  gas  in  the  dis- 
charge of  the  projectile,  and  upon  the  bottom  of  the  cham- 
ber, for  the  recoil  of  the  gun,  are  the  same  ;*  and  as  they 

*  In  reality,  and  in  the  ordinary  conditions  of  service,  the  weight  of  the 
charge  of  powder  being  £  to  \  that  of  the  ball,  we  cannot  make  this  suppo- 
sition, and  it  is  then  evident  that  the  force  of  the  gas  acting  against  the 
bottom  of  the  chamber  has  a  greater  tension  than  that  against  the  ball.  Con- 
sequently, the  velocity  of  recoil  is  greater  than  that  here  indicated. 


100  VIS   VIVA. 

are  exerted  during  the  same  time,  calling  P  and  P'  the 
weight  of  the  projectile  and  that  of  the  gun,  the  carriage 
included,  v  and  vf  the  elements  of  velocity  respectively 
imparted  in  an  element  of  time,  we  shall  have,  according 
to  the  proportion  offerees  to  velocities,  (No.  61,) 

F  :  P  :  :  v  :  gt,  and  F  :  P'  :  :  v'  :  gt ; 
whence  we  have 

P^^PV,  or  P  :  P7 :  :  v'  :  v, 

that  is  to  say,  that  the  velocities  imparted  in  the  element 
of  time  to  the  projectile  and  the  gun  are  in  the  inverse 
ratio  of  the  weights  of  these  bodies  and  as  the  total  veloci- 
ties imparted  at  the  moment  of  the  discharge,  are  equal 
to  the  sum  of  all  the  elements  of  velocity  which  they 
have  received  during  the  action  of  the  gas,  we  have  also 

P  :  P'  :  :  V  :  Y, 

Y  and  Y7  being  the  total  velocities  impressed  upon  the 
ball  and  gun. 

If  we  apply  this  consequence  to  the  infantry  percus- 
sion gun,  transformed  and  actually  in  service,  we  have 

P'=10.1555lb%     P=0.0639lb% 
whence 

P'    10.1555 


P       .0639 


=159. 


Now,  from  experiments  made  with  the  ballistic  pen- 
dulum, we  find  that  the  velocity  imparted  to  a  ball  16.37 
drachms,  by  a  charge  of  4.5  drachms  of  powder  is 

Y=1328.T6ft- 
We  deduce,  then,  from  the  above  proportion, 


, 


VIS  VIVA.  101 

This  velocity  is  quite  considerable,  and  we  would  re- 
mark that,  according  to  the  preceding  note,  the  real 
velocity  is  still  greater.  We  see  then  that  we  should  not 
attempt  to  lighten  portable  arms  beyond  a  certain  limit, 
if  we  do  not  wish  to  increase  the  velocity  of  recoil  in  too 
great  a  proportion.  From  the  preceding  values,  the  quan- 
tity of  motion  imparted  to  the  gun  would  be 


If  the  man  resists  the  recoil,  so  that  this  quantity  of 
motion  shall  be  spent  in  0.5",  for  example,  the  mean  effort 
exerted  at  the  shoulder  will  be 


To  diminish  this  effort,  it  is  best  to  interpose  between 
the  butt  and  shoulder  a  compressible  body  forming  a 
cushion.  Such  is  the  origin  of  the  epaulette. 

87.  Application  of  the  principle  of  Vis  Viva.  —  This 
principle  enables  us  to  appreciate  a  part  of  the  so  sudden 
effects  of  the  gas  of  powder  upon  fire-arms  and  projectiles. 
In  fact,  preserving  the  preceding  notations,  we  see  that 
the  Vis  Viva  imparted  to  a  projectile  is 


g, 

The  "  vis  viva  "  imparted  to  the  musket  is 

~V't=M/Y/i. 
& 

The  total  "  vis  viva  "  imparted  by  the  gas  is  then 
?V2  +  —  V/2=MV2+M'V/2. 


102  VIS    VIVA. 

But,  on  account  of  the  great  weight  of  the  gun  and  its 
stock,  the  vis  viva  imparted  to  the  projectile  is  much 
greater  than  that  impressed  upon  the  gun,  and  in  ordinary 
applications  the  latter  may  be  neglected. 

Thus  with  the  infantry  musket  we  have  for  the  pro- 
jectile 


1  QOQ  'Zfi^  _ 

g 

for  the  gun, 


The  ratio  of  their  vis  viva  is  equal  to 

PY2 
5^=159- 

The  quantity  of  work  developed  by  the  gas  of  the 
powder  upon  the  projectile  being  numerically  equal  to 
the  half  of  the  vis  viva  imparted  to  it,  we  have  actu- 
ally for  the  work  developed  by  0.018lbs-  of  powder 


and  we  see  that  in  the  comparison  of  mechanical  effects, 
or  of  the  quantities  of  work  produced  by  different  kinds 
of  powder,  we  may  be  satisfied  to  measure  them  by  the 
half  of  the  vis  viva  imparted  to  the  projectile. 

88.  Relation  between  the  charges  and  the  velocities. — 
The  work  of  the  gas  of  the  powder  should  evidently  be 
proportional  to  the  quantity,  and  thus  to  the  weight  of 
the  powder  producing  it,  so  long  as -we  can  admit  that  the 
charge  is  entirely  burnt  in  the  musket  before  the  dis- 
charge of  the  projectile  ;  which  is  sensibly  exact  for  mus- 
kets, even  with  more  than  the  common  charge,  but  is  not 


VIS    VIVA.  103 

so  for  cannons,  except  for  charges  of  i  to  £  the  weight  of 
the  ball. 

Consequently,  calling  C  and  Cx  the  charges,  P  and  P, 
the  weights  of  the  projectiles,  Y  and  Yt  the  velocities  im- 
parted to  them  by  these  charges,  we  should  have  the  pro- 
portion 

PY2  :  P;V7  :  :  C  :  C,, 

from  whence  we  conclude  : 

1st.  That  for  projectiles  of  the  same  weight,  or  for 
P=P,  we  have 

Y2  :  Y,2  :  :  C  :  Clf 

that  is  to  say,  with  the  same  gun  and  with  projectiles  of 
the  same  weight,  the  velocities  impressed  upon  the  latter 
are  to  each  other  as  the  square  root  of  the  charges. 

2d.  That  if  the  charges  are  equal,  or  C=Cn  we  have 

PVf =PlV1a ; 

which  shows  that,  with  fire-arms  of  the  same  proportions 
and  equal  charges,  the  velocities  of  the  projectiles  are  to 
each  other  in  the  inverse  ratio  of  the  square  roots  of  the 
weights  of  the  projectiles. 

89.  Verification  of  these  consequences  ~by  experiments. — 
The  first  of  these  laws,  enunciated  by  Hutton,  as  a  conse- 
quence of  his  experiments,  has  lately  been  the  object  of 
numerous  experiments  made  upon  guns  of  different  cal- 
ibre, and  balls  whose  windage  varied  between  extended 
limits.  Some  were  made  by  M.  Mallet,  Colonel  of  Artil- 
lery, with  common  musket  powder  ;  others  writh  powders 
of  different  kinds,  and  with  pyroxile  with  a  base  of 
cotton,  or  gun-cotton*  during  the  researches  ordered  by 
the  minister  of  war  to  be  made  upon  this  remarkable  sub- 
stance. 

The  results  of  these  experiments  are  entered  in  the 
following  table  : 


104 


VIS   VIVA. 


Cal.  of  Gun,  .0574ft. 
Diam.  of  Ball,  .0571  ft. 
Wt.  of  Ball,  .069  Ibs. 
Windage,  .0003ft. 


5O  t-  IO  »O  XQ  GO  r—  IO 
(O  «O  CO  CO  t^  IQ  CO  t-' 

OOO 


5ft. 
1  ft. 
Ibs 


Cal.  of  Gu 
Diam.  of  B 
Wt.  of  Ba 
Windage, 


COOOOOOOO 
-*  •*  00  O  <N:  b-  t&  tfl 


=" 


•VAIA  SIA 


ft. 
ft. 

Ibs. 


Cal.  of  Gnn, 
Diam.  of  Bal 
Wt.  of  Ball, 
Windage, 


t^QOOOOOO 
•^<MCOCOQOC1>—  l 


t^QOOOOOOOOOOOOOO 


OO 
COO 


•VAIA 


iOCOOCO'* 
i—  i  <M  O  •*  r-( 


•saixiooiaA 


Gun,  .057ft. 
of  Ball,  .053ft. 
Ball,  .057  Ib 
ge,  .004  ft. 


Cal. 
Dia 
Wt. 
Win 


OOOOOOOOO 
OiCO^OCOi-HTH^o 


•VAIA  SIA 


OiOJiMiOOOOt-COi-HOOCOb-OiO 


OJiMiOO 
'+l^-^lO 


•S3IU0013A 


1.  of  Gun,  .059  ft. 
am.  of  Ball,  .053  ft. 
t.  of  Ball,  .057  Ib 
indage,  .006 


COQOCOi-lOOOOOOOOOOO 


•VAIA  SIA 


O 
Oi 


VIS   VIVA. 


105 


To  free  these  results  from  anomalies  always  attending 
similar  researches,  however  carefully  conducted,  I  have 
presented  them  graphically  in  Figs.  40  and  41,  taking 
the  charges  for  the  abscissae,  and  the  vis  viva  for  the  or- 
dinates. 


In  the  first  experiments,  besides  the  charge  we  have 
varied  the  difference  in  diameter  of  the  gun  and  the  ball, 
or  what  is  termed  the  windage,  to  appreciate  the  influence 
of  this  quantity  upon  the  effect  of  the  powder. 


106 


VIS    VIVA. 


The  figures  show  the  exactness  of  the  law  of  propor- 
tionality between  the  charges  and  the  vis  viva :  indeed. 


Mean  Efforts. 


we  know  in  this  case,  that  the  curve  representing  the  re- 
lation between  these  two  quantities  must  be  a  straight  line 
passing  through  the  origin  of  co-ordinates. 

90.  Comparison  of  the  vis  viva  imparted  ~by  different 
powders. — Other  experiments  have  been  made  with  differ- 
ent kinds  of  powder,  and  with  pyroxile  or  gun-cotton. 


VIS   VIVA. 


107 


They  also  furnish  a  verification  of  this  important  law,  and 
have  led  to  the  following  formulae  between  the  vis  viva 
imparted  and  the  charges  of  powder.  The  guns  used 
were  in  calibre  0.057f%  and  the  balls  weighed  0.056lbs- 

Vis  viva  imparted  to  bullets  l)y  different  explosive 
materials. 


Charges  producing 

Kind  of  Material. 

Bullets. 

the  same 
ballistic  effect. 

Bouchet's  (  common,  coarse  for  blasting, 

MV2=  28.37  C 

.0323  Ibi. 

J  for  muskets 

MV2=  52.50  C 

.0176  " 

powucrs   i  Cinnon 

MV2=  59.00  C 

.0156 

Esquerdcs  |  Fine  sporting, 
powders  }  Extra  fine  sporting, 
Pyroxile  (carded)  from  Montreuil, 

MV2=  72.83  C 
MV2=  82.14  C 
MV2=159.25  C 

.0127 
.0100 
.0062 

Pyroxile  (carded)  Bouchet's, 
Pyroxile  (spun)  Bouchet's, 

MV2=142.00  C 
MV2=147.60  C 

.0065 
.0063 

These  results  show  how  much  the  effects  of  explosive 
materials  depend  upon  their  composition  and  mode  of 
preparation. 

91.  Use  of  the  consideration  of  the  mean  e forts. — In 

«/  «/  e*/ 

calculations  relative  to  the  proportions  to  be  given  to  dif- 
ferent parts  of  machines,  we  shall  frequently  substitute 
the  mean  efforts  for  the  variable  efforts  of  the  forces  act- 
ing upon  these  parts ;  but  to  indicate  by  an  example  all 
the  advantages  to  be  derived  from  a  consideration  of  the 
mean  efforts  in  appreciating,  at  least  approximately,  cer- 
tain very  complex  effects,  we  give  another  example  rela- 
tive to  what  transpires  in  the  combustion  of  explosive 
materials. 

92.  Comparison  of  the  effects  of  powder  and  of  pyrox- 
ile on  fire-arms. — When  the  discovery  of  a  process  by 
which  ligneous  substances  could  be  converted  into  ex- 
plosive materials  was  first  known,  the  rapid  combustion 
of  some  of  these  bodies,  especially  that  of  cotton  prepared 
in  concentrated  azotic  acid,  (first  called  powder-cotton, 
gun-cotton,  and  afterwards  pyroxile,)  seemed  to  many  to 
be  a  highly  valuable  property,  possessing  considerable 


108  vis  VIVA. 

advantage  over  common  powder.  But  experienced  artil- 
lery officers,  who  remembered  the  disastrous  effects  pro- 
duced upon  brass  cannons  by  powders  of  great  energy 
and  rapid  combustion,  for  the  introduction  of  which  into 
the  service  an  attempt  was  made  in  1828,  regarded,  on 
the  contrary,  this  property  as  more  dangerous  than  useful.. 
They  knew  that  the  inflammation  of  common  powder, 
though  gradual,  was  effected  so  rapidly  that  the  gas 
attained  its  maximum  tension  when  there  was  but  a  slight 
displacement  of  the  projectile. 

General  Piobert,  in  his  able  researches,  had  shown 
that  the  maximum  tension  was  established  more  quickly, 
as  the  powders  were  of  a  more  rapid  combustion,  and  to 
this  circumstance  he  attributed  the  speedy  destruction  of 
fire-arms  by  lively  and  dense  powders.  The  belief,  there- 
fore, was  well  founded,  that  gun-cotton,  by  reason  of  its 
rapid  combustion,  would  be  destructive  to  fire-arms. 

These  logical  deductions  from  the  known  facts  were 
by  no  means  acceptable  at  this  time  of  infatuation  for 
products  so  novel  and  extraordinary,  and  the  counsels  of 
prudence  were  attributed  to  the  prejudice  of  custom. 

The  best  means  of  deciding  the  question  was  by  a 
reference  to  experiments,  which  were  made  wTith  much 
care  and  diligence  on  a  great  scale.  The  course  adopted 
was  as  follows  : 

To  arrive  at  an  approximate  comparison  of  the  ten- 
sions of  the  gas  of  powder  and  of  pyroxile  at  different 
instants  of  the  motion  of  the  projectile  in  the  barrel,  there 
were  fired  in  succession,  with  charges  of  4.5  drachms  of 
war  powder,  and  1.69  drachms  of  pyroxile  with  cotton  for 
its  basis,  guns  of  a  calibre  0.059ft>,  whose  decreasing 
lengths  were  regulated  as  follows : 
3.55ft-  2.T3ft-  2.12ft-  1.62ft-  1.23ft-  O.S9ft-  0.61ft-  0.36ft- 

0.2Sft-  0.22ft-; 

which  corresponded  to  the  numbers  of  calibres  respect- 
ively equal  to 

64    49     38     29     22     16     11     7     5     4  calibres. 


VIS   VIVA.  109 

The  charges  of  4.5  drachms  of  powder  and  of  1.69 
drachms  of  pyroxile,  had  from  previous  experiments  been 
adopted  as  nearly  equivalent,  but  it  was  found  in  the  course 
of  the  experiments  that  1.6  drachms  of  pyroxile  sufficed 
to  impress  upon  the  same  ball  of  0.0635lbs>  weight  a  veloc- 
ity of  1235. 9ft>,  equal  to  that  imparted  by  4.5  drachms  of 
powder.  Future  comparisons  will  be  based  upon  these 
charges. 

The  velocities  imparted  to  balls  were  measured 
with  the  ballistic  pendulum,  by  placing  the  gun-barrel 
upon  a  frame,  so  that  the  face  of  the  muzzle  was  6.56ft> 
from  the  ballistic  receiver. 

Bearing  in  mind  that  the  vis  viva  imparted  to  the  ball 
is,  by  the  principle  of  vis  viva,  equal  to  double  the  quan- 
tity of  work  developed  by  the  gas,  and  that  the  mean 
effort  of  this  gas,  or  the  constant  effort  which  would  in 
each  case  impress  the  ball  with  the  same  vis  viva,  is  equal 
to  half  of  the  vis  viva  divided  by  the  length  of  path  de- 
scribed by  the  projectile  in  the  barrel,  we  see  that  from 
observation  of  the  velocity  of  the  latter,  which  is  termed 
the  initial  velocity,  we  may  easily  deduce  the  value  of 
the  mean  effort. 

It  is  also  evident,  that  the  value  so  determined  will 
always  be  below  the  maximum  effort,  and  will  decrease 
with  the.  length  of  the  barrel:  so  that  the  conclusions, 
from  a  comparison  of  the  mean  efforts  of  the  gas  of  pow- 
der and  of  pyroxile,  will  approach  more  nearly  the  truth, 
as  the  path  described  by  the  projectile  in  the  barrel  shall 
be  less,  and  will  very  nearly  approach  the  truth,  in  the 
first  moments  of  its  displacement,  which  are  precisely 
those  in  which  the  efforts  should  be  studied. 

The  length  of  the  chamber  occupied  by  the  charge 
was  the  same  for  the  powder  as  for  the  gun-cotton,  and 
was  0.157ft< ;  and  subtracting  this  from  the  interior  length 
of  the  barrel,  we  have  the  space  described  by  the  hind 
part  of  the  ball  in  the  barrel,  and  dividing  the  half  of  the 


110 


VIS    VIVA. 


imparted  vis  viva  by  this  length,  we  obtain  the  mean 
effort  sought. 

It  is  proper  to  remark  that  this  estimate  of  the  space 
described  by  the  ball,  while  subject  to  the  action  of  the 
gas,  is  that  usually  adopted  in  calculations  of  this  kind, 
but  it  is  not  wholly  exact.  In  fact,  when  the  centre  of 
the  ball  has  passed  the  face  of  the  muzzle,  a  portion  of  the 
gas  escapes  around  it ;  still,  these  gases  issuing  with  great 
velocity,  their  impulse  is  partly  continued  outwards. 
However  this  may  be,  the  value  abov  adopted  for  the 
space  described  by  the  ball  under  the  action  of  the  gas  is 
too  great  rather  than  too  small ;  consequently,  the  mean 
effects  deduced  from  it  are  too  small,  and  our  conclusions 
err  on  the  safe  side. 

We  have  represented  the  results  of  experiments  and 
of  calculations  in  two  different  ways.  In  the  first  (Fig. 
40)  we  have  taken  the  lengths  of  the  barrel  described  by 
the  ball  for  abscissae,  and  the  vis  viva  for  ordinates ;  in 
the  second  (Fig.  41)  we  have  also  taken  the  lengths  of  the 
barrel  described  by  the  ball  for  the  abscissa,  and  for  ordi- 
nates the  corresponding  mean  efforts  of  the  gas  of  powder 
and  of  pyroxile.  We  thus  have  a  graphic  expression  of 
results  contained  in  the  following  table : 

Results  of  comparative  experiments  upon  the  velocities, 
the  vis  viva,  and  the  mean  efforts  developed  by  the 
gas  of  war-powder  and  that  of  pyroxile. 


Lengths  of  barrel. 

Velocities  imparted. 

Vis  Viva  imparted. 

Mean  efforts  exerted. 

Total 

Described 

By  4.5  drms. 

By  1.61  drms 

By 

By 

By   gas 

By    gas 

by  the  ball. 

of  powder. 

of  pyroxile. 

powder. 

pyroxile. 

of  powder. 

of  pyroxile. 

a 

ft. 

ft. 

ft. 

Ibg.  ft. 

IbB.   ft. 

Ibs. 

Ibs. 

3.55 

3.39 

1235.9 

1235.5 

3015.1 

3013. 

447.4 

444.3 

2.73 

2.57 

1234.2 

1270.7 

3006.5 

3187.3 

584.9 

620.1 

2.11 

1.96 

1146.7 

1245.4 

2595.6 

3061.7 

662.1 

781. 

1.61 

1.46 

1039.6 

1176.2 

2133.2 

2730.8 

730.6 

935.2 

1.22 

1.06 

938.6 

1182.3 

1788.6 

2759.2     |      820. 

1301. 

0.89 

0.73 

856.9 

1348.6 

1449.5 

2264.9 

992. 

1541. 

0.61 

0.45 

724.9 

965.8 

1037.2 

1841.1 

1152. 

2045. 

0.39 

0.28 

530.4 

821.9 

555.2 

1333.6 

1207. 

2599. 

0.27 

0.12 

378.2 

577.3 

282.3 

657.6 

1176. 

2740. 

0.22 

0.06 

293.1 

341.2 

169.5 

302.7 

1412. 

'2521. 

VIS    VIVA.  Ill 

An  examination  of  Fig.  40  shows  : 

1st.  That  for  powder,  the  vis  viva  and  consequently 
the  velocity  of  the  ball  was  not  sensibly  increased  beyond 
A  length  of  barrel  2.62f%  answering  to  49  calibres. 

2d.  That  with  pyroxile,  the  vis  viva  and  maximum 
velocity  seems  to  correspond  with  the  same  length,  and 
to  decrease  with  greater  lengths. 

3d.  That  finally,  the  v-is  viva  imparted  by  charges  of 
4.5  drachms  of  powder,  and  of  1.6  drachms  of  pyroxile, 
are  equal,  for  lengths  of  3.55ft-  or  64  calibres,  but  that  for 
greater  lengths  the  pyroxile  lost  the  advantage  it  pos- 
sessed in  shorter  lengths. 

Fig.  41  shows  that  starting  with  a  length  of  3.55ft>,  for 
which  the  charge  of  4.5  drachms  of  powder  and  '1.6 
drachms  of  pyroxile  have  given  the  same  vis  viva,  and  so 
the  same  mean  effects,  the  effort  exerted  by  the  gas  of 
pyroxile  prevails  always  over  that  of  powder,  in  the  pro- 
portion of  the  diminution  of  the  length,  and  that  for  small 
lengths  of  barrel,  or  in  the  first  displacements  of  the  pro- 
jectile, the  mean  maximum  tension  of  the  gas  seems  to 
correspond  with  the  instant  of  0.246ft-  displacement,  and 

9ftfi9  4- 
was  then   2862.4lbs-   or  ^^=1044821lbs-*  per  sq.  ft, 

1044821 

or  finally,  —493.4  atmospheres,  while  the  mean 

-2116.4 

maximum  tension  of  the  gas  of  powder  was  not   over 

1  9QO  S 

1290.8lbs-  or  -±f^=471170lbs-  per  foot   square,  or 
.002  1  o9 

—222.6  atmospheres,  in  taking  even  its  value  an- 


swering  to  the  smallest  length,  which  seems  to  depart 
somewhat  from  the  law  followed  by  the  other  lengths. 

It  follows,  therefore,  that  the  mean  maximum  pressure 
produced  by  the  gas  of  powder  will  not  attain  the  half  of 

ft. 

059052 
*  The  surface  of  the  great  circle  of  the  ball  is  '  =.002739  •«>.  ft. 


112  VIS   VIVA. 

that  produced  by  the  gas  of  pyroxile,  for  charges  pro- 
ducing the  same  ballistic  effect. 

The  dimensions  of  infantry  muskets  are  such,  that 
when  the  projectile  is  displaced  0.246ft-  it  is  found  in  a 
part  of  the  barrel  having  a  thickness  T—  0.017716ft>,  and 
it  is  readily  seen,  from  the  formula  of  the  resistance  of  a 
cylinder  to  rupture,  supposing  the  rnetal  to  be  of  a  me- 
dium quality,  that  the  interior  pressure  capable  of  pro- 
ducing rupture  will  be  for  the  unit  of  surface 

2T       E        .03543  8195580 


and  when  the  metal  is  impaired  by  the  firing,  or  is  of  an 
inferior  quality,  it  is 

.03543  5122210 


Thus,  in  th'e  last  case,  the  maximum  pressure  of  the 

222  5       1 
gas  of  the  powder  would  only  be         '—  =  -—  of  the  mod- 

ulus of  rupture,  while  that  of  pyroxile  would  be 
493.4        1 


1451.4    2.94 

If  we  refer  to  the  comparative  results  previously  re- 
ported upon  the  vis  viva  imparted  by  explosive  materials, 
according  to  which  we  have  seen  that  the  charge  of 
pyroxile  was  to  the  equivalent  charge  of  fine  sporting 
powder  as  72.83  :  159.25,  we  see  that  the  charge  of  pyrox- 
ile equivalent  to  that  of  15.5  drachms  of  sporting  powder 
used  in  the  tests  of  guns,  would  be  7  drachms.  ISTow  it 
sometimes  happens  that  guns  burst  with  a  charge  of  15.5 
drachms  of  sporting  powder,  and  since,  with  the  same 
ballistic  effect,  the  pyroxile  develops,  at  the  first  instant, 
much  greater  pressure  than  the  powder,  it  would  seem  to 


VIS   VIVA.  113 

follow  that  guns  could  not  generally  resist  a  charge  of 
6.77  drachms  of  pyroxile. 

Without  attaching  an  undue  amount  of  precision  to 
these  calculations,  we  may  yet  have  in  them  a  confirma- 
tion of  the  fears  first  entertained  as  to  the  effects  of  the 
rapid  combustion  of  pyroxile. 

Later  experiments  have  confirmed  these  deductions, 
and  when  it  was  wished  to  determine  the  velocities  im- 
parted by  increased  charges,  it  was  constantly  the  case 
that  new  guns  burst  with  charges  of  3.95  to  4.23  drachms, 
and  sometimes  with  less. 

If  we  bear  in  mind  that  infantry  guns  are  made  of  a 
choice  quality  of  iron,  submitted  to  a  close  inspection  and 
even  severe  tests  before  reception,  and  that  the  thickness 
of  the  metal  is  much  greater  than  that  of  fowling-pieces, 
we  cannot  doubt  that  the  use  of  pyroxile  in  portable  fire- 
arms is  far  from  affording  the  same  security  as  powder. 

93.  Consumption  and  restoration  of  work  ly  inertia. — 
It  follows,  from  the  above,  that  to  impart  to  a  body  a 
certain  velocity,  answering  to  a  certain  vis  viva,  a  quan- 
tity of  work  must  be  developed  which  is  expressed  by  the 
half  of  the  vis  viva,  and  reciprocally,  if  the  body  lose  a 
part  or  the  whole  of  its  vis  viva,  a  work  is  developed  in 
virtue  of  its  inertia,  which  is  also  expressed  by  half  of  the 
vis  viva  destroyed. 

In  the  first  case  the  motive  work  is  transformed  into 
an  imparted  vis  viva,  and  in  the  second  the  vis  viva  is 
transformed  into  a  resistant  work. 

94.  Rams  for  pile-driving,  punching  machines,  c&c. — 
In  driving  piles,  the  work  employed  to  raise  to  a  height 
H,  a  ram  of  the  weight  P,  is  transformed  in  its  descent 

P  Y2 
into  a  vis  viva  -    —— PH ;  for  when  the  ram  reaches  the 

9  * 

head  of  the  pile,  it  develops  by  its  inertia  efforts  which 
8 


114  VIS   VIVA. 

compress  the  head,  overcoming  its  resistance,  sinking,  and 
so  producing  a  corresponding  useful  work. 

In  boring  and  punching  metals  with  a  ram,  the  resist- 
ance overcome  is  that  opposed  by  the  metal  to  the  sepa- 
ration of  its  molecules,  and  the  thickness  of  the  piece  is 
the  space  described. 

The  example  already  cited,  of  the  action  of  powder 
upon  balls,  shows  us,  at  first,  the  work  transformed  into  a 
vis  viva,  then,  during  the  penetration  of  the  balls  into 
any  medium,  the  vis  viva  is  employed  in  overcoming  the 
resistance  of  the  medium. 

96.  Work  expended  during  the  period  of  compression 
from  the  shock  of  two  non-elastic  lodies.  —  Calling  M  and 
Y  the  mass  and  velocity  of  the  impinging  body,  and  M' 
and  Y'  the  mass  and  velocity  of  the  body  shocked,  we 
would  remark  that  the  total  vis  viva  of  these  two  bodies 
before  the  shock  was  MY2+M'Y/2,  and  that  after  the 
shock,  since  they  move  with  a  common  velocity, 

TT_MY+M'Y' 

"M+M7"' 

their  vis  viva  will  be  (M+M')!!2.  Consequently,  the  vis 
viva  destroyed  during  the  compression,  and  employed  in 
producing  it  will  be 


and  the  work  consumed  by  this  compression  is 

MW_ 
fA 


If  the  body  shocked  had  before  the  shock  a  motion 
against  the  impinging  body,  we  have  seen  when  the  body 


VIS  VIVA.  115 

M'  after  the  shock  recedes,  and  takes  the  direction  of  the 
body  M,  that  we  have  for  the  common  velocity  after  the 
shock 

TT_MV-M/Y/ 

M+M7"' 
and  then  the  loss  of  vis  viva  producing  the  compression  is 


and  the  work  consumed  by  this  compression  is 


If,  after  the  shock  in  the  last  case,  the  velocity  II  were 
zero,  which  happens  when  MY^M'Y',  the  work  con- 
sumed by  the  compression  is  reduced  to  ^(MY'+M'Y'2), 
which  is  quite  evident,  since  the  two  bodies  are  brought 
to  rest  by  the  shock. 

If  the  mass  of  the  impinging  body  is  very  great  com- 
pared with  that  of  the  body  shocked,  the  loss  of  work 


M 

jyr/ 

is  reduced  by  reason  of  the  smallness  of  the  ratio  -^  to 

iM'(V=FV')*;  in  the  first  case  iM'(Y-Y')2  is  the  work 
answering  to  the  vis  viva  gained  by  the  body  shocked, 
and  in  the  second  £M'(Y-f  Y')2  is  the  work  answering  to 
the  vis  viva,  due  to  the  sum  of  the  velocity  Y'  which  the 
body  shocked  has  lost  in  one  direction,  and  of  Y  which 
it  has  received  in  an  opposite  direction,  because  then  U 
is  reduced  to 


- 
MV-M'V'_     _M__V 

M+M'  M'          ' 


116  vis  VIVA. 

in  consideration  of  the  smallness  of  —  in  its  ratio   with 

M 
unity. 

If  the  body  shocked  were  at  rest  before  the  shock,  we 
have  V=0,  and  the  loss  of  work  due  to  the  compression  is 


so  long  as  the  velocity  of  the  impinging  body  is  not  sen- 
sibly altered,  and  U=V  as  above. 

If,  on  the  other  hand,  the  mass  of  the  body  shocked  is 
very  great  compared  with  that  of  the  impinging  body, 
we  have  for  the  loss  of  work  relative  to  the  first  case, 
where  the  bodies  move  in  the  same  direction, 


on  account  of  the  smallness  of  the  ratio  =-=•. 

M 

97.  Of  the  work  due  to  compression,  and  the  return  to 
the  primitive  form  in  the  case  of  elastic  bodies.  —  If  the 
bodies  are  perfectly  elastic  in  their  return  to  the  primi- 
tive form,  the  molecular  springs  must  develop  the  same 
efforts  in  returning  to  the  same  degrees  of  tension,  and 
the  points  of  application  describing  the  same  paths  as  in 
the  compression,  the  total  work  developed  by  these  efforts, 
varying  in  the  same  manner  in  both  cases,  will  be  the 
same,  and  the  work  due  to  the  unbending  of  the  molecular 
springs  will  be  the  same  as  the  work  consumed  in  their 
compression  ;  so  that  in  reality  the  consumption  of  the 
work  due  to  the  shock  is  nothing. 

98.  The  work  lost  in  the  shock  of  bodies  imperfectly 
elastic.  —  If  the  bodies  are  imperfectly  elastic,  as  is  gener- 
ally the  case  ;  or  rather,  if  the  flexures  and  changes  they 
experience  during  the  shock  exceed  the  limits  of  those 


VIS   VIVA.  117 

which  can  be  produced  without  an  alteration  of  the  elas- 
ticity, then  the  parts  shocked  remain  more  or  less  changed 
in  form,  and  only  a  part  of  the  work  consumed  in  pro- 
ducing it  is  restored.  There  is  then  a  loss  of  work. 

Now,  in  machines  imparting  shocks  it  nearly  always 
happens,  either  in  their  first  use,  or  after  a  period  of  ser- 
vice, that  the  elasticity  of  the  parts  in  contact  is  more  or 
less  changed,  and  that  the  loss  of  work  by  the  shock  is 
very  nearly  the  same  as  that  which  takes  place  in  the 
shock  of  soft  bodies.  This  last  quantity  is,  moreover,  the 
greatest  limit  which  this  loss  can  attain. 

In  recapitulating,  we  see  that  in  shocks  there  is  in 
practice  always  a  more  or  less  great  loss  of  work,  due  to 
the  disturbance  of  the  parts  in  contact,  and  that  it  is  well 
to  substitute,  as  far  as  possible,  parts  with  a  continuous 
motion,  for  those  working  with  shocks,  intermissions,  or 
sudden  changes  of  velocity. 

99.  Masses  in  motion  may  be  regarded  as  reservoirs  of 
work. — It  follows,  from  the  preceding  remarks,  that  bodies, 
in  virtue  of  their  inertia,  absorb  and  store  up  mechanical 
work,  when  the  forces  are  employed  in  communicating 
to  them  velocity  and  vis  viva,  and  on  the  contrary,  trans- 
mit and  restore  the  work  when  their  motion  is  retarded. 
In  this  view  we  may  regard  them  as  reservoirs  of  mechan- 
ical work,  which  are  filled  during  acceleration,  and  emp- 
tied in  the  retardation,  absolutely  in  the  same  manner  as 
the  reservoirs  of  hydraulic  motors. 

100.  EXAMPLE. — We  have  already  seen,  in  Art.  94,  that 
it  was  in  virtue  of  the  work  thus  stored  up  that  the  pile- 
driving  rams  produced  their  effects ;  it  will  be  the  same 
whatever  the  number  of  intermediate  parts  of  the  ma- 
chine:  we  find  a  striking  example  of  a  similar  application 
in  the  walking-beam  employed  in  many  mechanical  de- 
partments. 


118  VIS   VIVA. 

If  the  fly-wheel  of  a  machine  is  set  in  motion  with  a 
certain  velocity,  by  any  motive  force,  and  then  left  to 
itself,  it  will  continue  to  move  until  the  frictions  and  other 
resistances  have  entirely  expended  the  work  which  was 
accumulated  under  the  action  of  the  motor — when  this 
work  is  consumed  it  will  stop. 

But  if,  while  animated  with  a  certain  velocity  testify- 
ing to  the  accumulation  of  a  certain  work,  we  oppose  to 
the  machine  a  useful  resistance,  we  see  that  it  then  devel- 
ops a  useful  mechanical  action,  such,  for  example,  as  the 
coining  of  money,  the  stamping  a  metal  plate  into  a  given 
form,  the  piercing  of  holes,  &c. 

101.  Periodical  motion. — If  the  motion  of  the  body 
varies  periodically,  that  is  to  say,  if  its  velocity  increases 
or  decreases  successively  in  equal  quantities,  it  is  evident 
that  the  work  consumed  in  the  period  of  acceleration  is 
equal  to  the  resistant  work  during  retardation,  and  that 
then  the  total  work  developed  by  inertia  is  nothing.  If 
we  watch  what  passes  in  these  successive  periods,  where 
the  velocity  and  vis  viva  become  without  ceasing  the 
same,  at  the  end  of  each  period,  it  is  not  necessary  to  take 
any  account  of  the  vis  viva. 

We  shall  see  hereafter  the  great  importance  of  inertia 
in  the  work  of  machines. 


COMPOSITION  OF  MOTIONS,  VELOCITIES,  AND 
FOKCES. 

102.  Composition  and  resolution  of  simultaneous  mo- 
tions. — We  have  thus  far  considered  material  points  as 
animated  by  a  single  motion,  or  solicited  by  a  single  force, 
and  before  extending  the  preceding  theorems,  it  would  be 
well  to  examine  what  passes  when  a  body  or  material 
point  is  impressed  simultaneously  with  many  motions,  or 
solicited  by  many  forces. 

Observation  affords  frequent  examples  of  its  occur- 
rence. Thus,  when  a  traveller  promenades  the  deck  of  a 
steamboat  under  way,  he  is  impressed  with  the  motion 
of  the  vessel  as  well  as  that  of  his  own  will.  If,  while 
walking,  he  throws  a  body  from  him,  this  body  already 
partaking  of  the  two  motions  of  the  traveller,  takes  a  third 
in  falling  upon  the  deck ;  besides,  the  vessel  partakes  of 
the  daily  motion  of  rotation  of  the  earth,  and  also  of  its 
motion  about  the  sun. 

All  these  motions  are  simultaneous  and  are  independ- 
ent of  each  other,  since  the  causes  which  produce  them 
are. 

By  a  very  simple  experiment  of  M.  Tresca,  sub-director 
of  the  Conservatoire,  this  independence  of  simultaneous 
motion  is  made  very  apparent.  In  placing  the  chronom- 
etric  cylindrical  apparatus,  described  in  No.  80,  upon  a 
truck  impressed  with  a  uniform,  or  even  variable  motion, 
and  in  repeating  during  this  motion  the  experiment  of  the 


120      COMPOSITION   OF  MOTIONS,   VELOCITIES,   AND  FORCES. 

fall  of  bodies,  left  to  the  action  of  gravity,  it  was  seen  that 
the  parabola  traced  by  the  style  was  exactly  the  same  as 
was  obtained  when  the  apparatus  was  immovable.  The 
vertical  motion  of  the  heavy  body,  and  the  rotary  motion 
of  the  cylinder,  were  wholly  independent  of  the  motion 
of  the  apparatus. 

From  this  principle  of  the  independence  of  simultane- 
ous motions  follow  rules  which  enable  us  to  determine  the 
real  motion  resulting  from  many  simultaneous  known 
motions,  and  which  is  called  the  resultant  motion. 

103.  Case  of  the  simultaneous  motions  having  the  same 
direction. — The  first  and  most  self-evident  case,  is  that  of 
a  material  point  impressed  with  simultaneous  motions 
acting  in  the  same  straight  line.  They  are  added  if  in 
the  same  direction,  and  subtracted  if  in  opposite  direc- 
tions, in  order  to  obtain  the  resultant. 

In  the  case  of  a  person  walking  upon  the  deck  of  a 
vessel,  it  is  evident  that  if  he  walks  in  the  direction  the 
vessel  is  going,  his  motion  and  displacement  in  respect  to 
a  fixed  point  on  the  shore,  supposed  to  be  parallel  to  these 
motions,  will  be  equal  to  the  sum  of  the  displacements  of 
the  vessel  and  of  his  path  described  upon  the  deck.  If 
the  boat  has  advanced  26  feet  while  the  traveller  has 
passed  9  feet  forward,  the  displacement  of  the  traveller  in 
respect  to  the  banks  will  be  26ft-+9ft—35ft- 

If  he  walked  15  feet  in  an  opposite  direction  to  the 
boat,  while  the  boat  advances  26  feet,  his  displacement, 
or  the  space  described  by  him  in  respect  to  the  banks, 
will  then  be 

26ft--15ft-=llft- 

If  the  traveller  walks  towards  the  stern  of  the  vessel  a 
distance  equal  to  that  which  the  latter  has  advanced  in 
the  same  time,  his  displacement  in  respect  to  the  banks  is 
nothing,  and  though  impressed  with  two  simultaneous 
motions,  he  is  at  rest  in  respect  to  the  banks. 


COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND   FORCES.       121 

Finally,  if  the  traveller  walks  towards  the  stern  a  dis- 
tance of  30  feet,  or  4  feet  greater  than  that  described  by 
the  boat,  his  displacement  in  respect  to  the  banks  will  be 
negative,  an  expression  indicating  that  he  has  receded  in- 
stead of  advancing  in  respect  to  the  banks. 

It  would  be  the  same  for  any  number  of  simultaneous 
motions  directed  in  the  same  straight  line  ;  calling  : 

S,  S',  S",  &c.,  the  paths  directed  from  left  to  right, 
and  regarding  them  as  positive  : 

Si,  S/,  S/',  &c.,  the  paths  directed  from  right  to 
left,  and  regarding  them  as  negative  or  subtractive,  the 
total  resultant  path  of  these  simultaneous  motions  will  be 
equal  to 

,-^-^/—  S/'—  &c., 


which  is  expressed  in  saying  that  the  resultant  path  is  the 
algebraic  sum  of  all  the  simultaneous  or  component  paths  : 
understanding  here  by  the  word  sum  the  result  of  the 
operation  of  adding  all  the  paths  from  left  to  right,  and 
subtracting  all  those  from  right  to  left. 

104.  Composition  of  several  simultaneous  velocities  di- 
rected in  the  same  straight  line.  —  All  that  we  have  said 
upon  the  composition  of  spaces  simultaneously  described 
by  a  material  point  in  the  same  direction,  applies  to  the 
simultaneous  velocities  impressed  upon  a  point,  since  in 
uniform  motion  the  velocities   are  proportional  to   the 
spaces  described  in  the  same  time,  and  since  in  variable 
motions  the  velocities  are  those  of  uniform  motions  which 
the  bodies  would  possess  if  these  motions  ceased  to  be 
variable. 

105.  Composition  of  two  motions  directed  in  any  man- 
ner. —  Let  us  consider  now  the  point  A,  which  may  be  the 
point  of  a  pencil  placed  upon  the  inclined  rulers  MA1ST. 
If  the  rulers  are  moved  uniformly  a  quantity  AB,  its  side 
AM  will  be  displaced  parallel  to  itself  the  same  quantity, 


122      COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND  FORCES. 

moving  also  uniformly,  and  with  it  the  point  of  the  pen- 
cil to  which  it  is  attached.     But  if  in  the  same  time  T, 

the  pencil  moves  upon  the 
side  AM,  uniformly  a  quan- 
tity AD,  it  is  easy  to  see  that 
at  the  end  of  the  time  T,  the 
point  of  the  pencil  will  have 
arrived  at  the  point  C,  and  at 
the  summit  of  the  parallelo- 
gram constructed  upon  AB  and  AD  as  sides. 

In  fact,  this  point  constantly  resting  upon  the  side  AD 
being  displaced  with  it  parallel  to  its  primitive  position  a 
quantity  equal  to  AB,  would  be  found  upon  the  line 
BC,  parallel  to  AD,  and  as  it  is  also  displaced  in  the 
direction  AM,  by  the  quantity  AD,  it  would  likewise  be 
found  upon  the  line  DC,  drawn  parallel  to  AB.  The  in- 
tersection of  the  two  lines  BC  and  DC  determines  the 
direction  of  the  diagonal  of  the  parallelogram  constructed 
upon  the  simultaneous  paths. 

Whence  it  follows  that,  when  a  material  point  is  ani- 
mated ~by  two  simultaneous  motions  in  two  given  direc- 
tions, the  position  of  the  point  at  the  end  of  these  two 
motions  will  be  at  the  extremity  of  the  diagonal  of 
the  parallelogram  constructed  upon  the  two  paths  as 
sides. 

The  distance  AC,  at  which  the  point  is  found  from  its 
first  position  A,  is  called  the  resultant  path,  and  the  two 
simultaneous  paths  AB  and  AD  are  called  the  component 
paths  or  the  relative  paths  passed  over  in  the  direction  of 
the  lines  AN  and  AM. 

For  any  two  other  but  also  simultaneous  paths  AB' 
and  AD'  passed  over  by  the  point  A,  in  another  time  T', 
the  point  A  will  arrive  at  a  position  A'  determined  by  the 
extremity  of  the  diagonal  AC'  of  the  parallelogram  con- 
structed upon  the  paths  AB7  and  AD'. 

!N~ow,  as  these  second  simultaneous  paths  are  by  hy- 


123 

pothesis  described  with  a  uniform  motion  as  well  as  the 
first,  we  have 

AB  :  AB'  :  :  T  :  T 

and 

AD  :  AD' :  :  T  :  T, 
whence 

AB  :  AB' :  :  AD  :  AW. 

The  angles  at  A  being  moreover  equal,  it  follows  that 
the  triangles  ABC  and  AB'C',  ADC  and  AD'C'  are  simi- 
lar, and  that  the  diagonals  AC  and  AC'  are  in  the  same 
direction. 

Moreover,  the  diagonals  AC  and  AC7  are  also  propor- 
tional to  the  times  T  and  T'  employed  in  reaching  the 
points  C  and  C'. 

When  a  material  point  moves  simultaneously  and  uni- 
formly in  two  given  directions,  the  path  really  described 
in  virtue  of  these  two  motions,  and  called  the  resultant 
path,  is  represented  in  magnitude  and  direction  ~by  the 
diagonal  of  a  parallelogram  constructed  upon  the  two 
paths  simultaneously  described,  and  its  motion  in  this 
direction  is  uniform  with  a  velocity  represented  by  the  ra- 

,.      AC    AC7 
/o/)          — • 

~f  ~~  T'' 

The  first  proposition  of  No.  105  enables  us  to  determine 
the  position  of  the  point  in  consequence  of  its  two  simul- 
taneous displacements,  the  second  gives  us  its  real  motion. 

Reciprocally,  when  a  material  point  moves  in  a  right 
line  AC,  uniformly  or  not,  we  may  always  find  its  simul- 
taneous displacements,  as  referred  to  any  two  given  direc- 
tions. It  suffices  for  this  to  construct  a  parallelogram 
whose  diagonal  is  AC,  and  whose  sides  AB  and  AC  are 
parallel  to  the  given  directions. 

We  see  that  a  path  or  a  given  motion  may  then  be 
resolved  or  decomposed  in  an  infinity  of  ways  into  two 
others  with  given  directions,  so  that  the  two  paths  or  mo- 
tions shall  answer  to  but  one  path  or  resultant  motion. 


124:      COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND  FORCES. 


O 


We  might  demonstrate  also  that  if  the  two  simulta- 
neous motions  of  the  point  A  in  the  directions  AIST  and 
AM  are  uniformly  accelerated,  the  resultant  motion  along 
the  diagonal  AC  will  be  so  likewise. 

106.  Variable,  motion. — All  that  has  been,  said  being 
independent  of  the  absolute  magnitude  of  the  paths  and 
velocities,  will  hold  good  for  two  infinitely  small  compo- 
nent paths. 

Thus  in  curvilinear  motion, 
an  element  of  the  infinitely  small 
path  AC  may  be  decomposed 
into  two  other  infinitely  small 
paths  described  parallel  to  any 
two  given  axes  in  the  same 
plane;  and  reciprocally,  if  we 
know  the  relative  elementary 
paths  AB  and  AD  described  in 
an  element  of  time  in  the  direc- 
tion of  the  axes  Ox  and  Oy,  we 
may  deduce  from  them  the  absolute  elementary  path  AC 
described  by  the  body. 

We  would  remark  that  this  absolute  elementary  path 
AC  is  the  element  of  a  curve,  whose  prolongation  gives 
the  tangent  AT,  at  the  point  A,  and  as  its  direction  de- 
pends upon  the  ratio  of  the  relative  paths  AB  and  AD, 
and  not  upon  their  magnitude,  it  follows,  that  if  this  ratio 
is  known  we  can  determine  this  diagonal  or  tangent  by 
constructing  upon  the  directions  of  AB  and  AD,  a  paral- 
lelogram whose  sides  have  to  each  other  the  same  ratio, 
and  tracing  its  diagonal.  This  principle  is  often  advan- 
tageously applied  in  tracing  tangents  of  curves. 

Moreover,  in  variable  motion,  we  see,  if  the  ratio  of 
the  elementary  path  AC  to  the  element  of  the  time  t  em- 
ployed in  describing  it,  is  given,  the  construction  of  the 

parallelogram  ABCD  will  give  the  ratios  -  — ,  — ,  or  the 

t       t 


FIG.  43. 


125 


relative  paths  AB  and  AD,  described  in  the  same  time, 
which  will  be  the  values  of  the  relative  velocities  in  the 
direction  of  the  axes,  and  that  if  the  absolute  velocity  is 
proportional  to  AC,  the  relative  velocities  will  be  propor- 
tional to  AB  and  AD. 

Then,  in  variable  motion,  the  velocity  at  any  instant 
can  be  decomposed  into  two  others,  in  any  two  given  direc- 
tions, and  represented  in  magnitude  l>y  the  sides  of  a  paral- 
lelogram constructed  upon  this  velocity  and  the  diagonal. 

Reciprocally,  the  resultant  velocity  is  the  diagonal  of 
a  parallelogram  constructed  upon  the  relative  velocities. 

107.  Case  where  the 
directions  of  the  compo- 
nents are  at  right  angles. 
— In  this  case  the  paral- 
lelogram is  a  rectangle, 
the  diagonal  the  hypothe- 
nuse  of  a  right  angled 
triangle,  and  its  square  is  equal  to  the  sum  of  the  squares 
of  the  sides.  We  have,  then,  the  simple  relation : 


=V  cos  CAD. 


Y'=V.:|?=VcosCAB, 


108.  Composition  of  any  number  of  simultaneous  mo- 
tions or  velocities  in  the  same  plane.  —  We  see  by  the  pre- 
ceding that  the  path  or  resultant  velocity  of  two  simulta- 
neous motions  in  any  two  directions  will  be  determined 
in  constructing  the  triangle  ABC,  and  drawing  the  side 
AC,  and  taking  in  the  given  directions  AB  and  BC=AD 
respectively  equal  to  the  spaces,  or  the  relative  and  simul- 
taneous velocities.  If  the  body  is  also  impressed  with  a 
third  motion,  or  a  third  velocity  AE,  we  construct  the 
triangle  ACF,  in  which  AC  is  the  motion  or  resultant 


126      COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND   FOECES. 


velocity  of  the  two  preceding,  and  OF  is  equal  and  par- 
allel to  AE.  AF  will  consequently  be  the  motion  and 

resultant    velocity 
jgr  of  the  two  simulta- 

neous motions  AC 
and  AE,  or  the 
three  motions  or 
velocities  AB,  AD 
and  AE.  So  for  a 
fourth  motion  or 
velocity  AG,  the 
motion  or  resultant 
velocity  will  be  giv- 
en by  the  side  AH 
of  the  triangle  AFH 
in  which  AF  is  the 
preceding  resultant, 
and  FH  is  equal  and  parallel  to  AG.  Then,  in  general, 
the  motion  or  the  resultant  velocity  of  many  simultaneous 
motions  or  velocities  in  the  same  plane  will  be  given  in 
magnitude  and  direction  by  the  last  side  of  the  polygon 
ABCFH,  &c.,  constructed  from  the  origin  A,  with  sides 
equal  and  parallel  to  the  given  simultaneous  motions  or 
velocities. 

If  we  project  the  last  side  of  the  polygon  thus  con- 
structed, upon  any  line,  by  perpendiculars  or  by  parallel 
lines  in  any  direction,  a  simple  inspection  of  the  figure 
shows  that 

A'H'^A'B'+B'C'+C'F'-F'H',  &c., 

which  signifies  that  the  projection  of  the  last  side,  or  the 
resultant  path  or  velocity,  is  equal  to  the  algebraic  sum  of 
the  projections  of  the  sides,  or  simultaneous  paths  orveloci- 


We  understand  here  by  the  algebraic  sum,  the  result 
obtained  by  adding  or  taking  as  positive,  the  sides,  paths, 
or  velocities,  in  the  real  direction  of  the  motion,  and  by 


COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND  FOKCES.      127 

subtracting  or  taking  negatively  the  sides,  paths  or  veloci- 
ties in  the  opposite  direction. 

It  follows  from  this,  that  if  the  last  side  is  zero,  and 
the  polygon  returns  upon  itself,  the  resultant  path  or 
velocity  is  zero,  and  the  body  is  not  displaced,  and  has  no 
velocity,  notwithstanding  the  relative  motions  imparted 
to  it.  It  is  also  the  case,  when  the  algebraic  sum  of  the 
paths  or  the  velocities  projected  upon  the  same  straight 
line  is  zero. 


109.  Resultant  of  three  simultaneous  motions  or  veloci- 
ties in  space. — If  the  body  is  impressed  with  three  simul- 
taneous motions  or  velocities  AB,  AD,  AF,  in  any  three 
directions  in  space,  it  is  evident  that  if  we  at  first  com- 
pound AB  and  AD,  then  their  resultant  AC  with  AF,  or 
AB  and  AF,  and  their  resultant  AE  with  AD,  or  AD  and 
AF  and  their  resultant 
AG  with  AB,  we  shall 
find  in  all  cases  for  the 
final  resultant  the  diag- 
onal AH  of  the  paral- 
lelopipedon  constructed 
upon  the  given  motions 
or  velocities. 

Then,  the  resultant 
of  three  simultaneous 
motions  or  velocities  in 
space,  is  represented  in 
magnitude  and  direction 
lyy  the  diagonal  of  the 
parallelopipedon  con- 
structed upon  these  three'  FIG  4G 
motions. 


110.  Reciprocally,  any  motion  or  velocity  may  he  de- 
composed into  three  motions  or  velocities  according  with 


128      COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND   FORCES. 

three  given  directions. — Let  AH  be  the  path  described  or 
the  velocity :  we  may  decompose  it  into  two  others,  one 
according  with  one  of  the  given  directions,  the  other 
following  AC  in  the  plane  of  the  other  two  directions, 
and  regard  the  body  as  impressed  with  these  two  simul- 
taneous motions  or  velocities.  Then  we  may  decompose 
the  motion  or  velocity  AC  into  two  others  AB  and  AD, 
according  with  the  two  other  given  directions. 

The  motion  or  velocity  All  will  then  be  replaced  by 
the  three  motions  or  velocities  AF,  AB,  and  AX),  in  the 
three  given  directions. 

Case  where  the  components  are  at  right  angles. — If  the 
three  directions  are  at  right  angles,  putting 

AB=Y',    AD=Y",    AF=V",  and  AH=Y, 
we  have 

Y=  v/Y/2+Y//2+v//a, 

and 
Y'=YcosBAH,  Y"=YcosDAH,  V'"=VcosFAH. 

111.  Resultant  of  any  nuinber  of  simultaneous  motions 
or  velocities. — If,  instead  of  being  impressed  with  three 
simultaneous  motions  or  velocities  AB,  AD,  AF,  the. body 
had  a  fourth,  it  is  readily  seen,  that  the  final  motion  or 
velocity  would  be  represented  in  magnitude  and  direction 
by  the  diagonal  of  the  parallelogram  constructed  upon 
the  resultant  of  the  first  three  motions,  and  upon  the 
fourth  as  sides ;  now,  this  line  is  the  last  side  of  the  poly- 
hedron formed  on  the  supposition,  that  the  body  receives 
these  simultaneous  motions  or  velocities. 

Then,  in  general,  the  resultant  'motion  or  the  velocity 
of  any  number  of  simultaneous  motions  or  velocities,  di- 
rected in  any  manner  in  space,  is  represented  in  magni- 
tude and  direction  ~by  the  last  side  of  the  polyhedric  poly- 
gon, formed  on  the  supposition  that  the  body  was  succes- 
sively impressed  with  these  simultaneous  motions. 


COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FORCES.       129 

But  we  arrive  more  simply  at  the  determination  of 
the  motion  or  resultant  velocity  by  recalling  our  previous 
statement,  that  any  motion  of  translation  may  be  decom- 
posed into  three  other  simultaneous  motions,  in  any  three 
given  directions,  which  are  the  sides  of  a  parallelopipedon? 
whose  diagonal  is  the  motion,  and  whose  sides  follow  the 
given  directions. 

This  being  established,  if  we  conceive  each  of  the  mo- 
tions, or  each  of  the  simultaneous  velocities,  impressed 
upon  the  body  to  be  thus  decomposed,  the  motion  or  the 
final  velocity  will  not  be  altered.  But  as  all  motions  or 
velocities  along  the  same  axes  have  partial  resultants 
equal  to  the  sum  of  the  components,  in  these  directions,  it 
follows  that  the  movement  or  resultant  velocity  will  be 
represented  in  magnitude  and  direction  ~by  the  diagonal 
of  the  parallelopiped  constructed  upon  the  sums  of  compo- 
nents of  partial  motions  in  any  three  directions. 

Following,  then,  the  reasoning  of  Art.  108,  and  sup- 
posing that  after  having  compounded  into  a  single  motion 
all  the  simultaneous  motions  impressed  upon  the  same 
material  point,  we  project  these  motions,  or  the  resultant 
motion  or  the  corresponding  velocities  upon  any  axis,  by  as 
many  planes  perpendicular  to  this  axis,  we  shall  see  that  the 
projection  of  the  resultant  motion  or  velocity,  which  is 
the  diagonal  of  the  polygon  already  mentioned,  is  equal 
to  the  algebraic  sum  of  projections  of  the  component  mo- 
tions or  velocities. 

112.  Case  where  the  resultant  is  zero. — When  the  line 
joining  the  extremities  of  the  first  and  last  side  of  the 
plane  or  polyhedric  polygon,  formed  upon  the  directions 
of  the  component  paths  or  velocities  is  zero,  which  hap- 
pens when  the  polygon  returns  upon  itself,  the  resultant 
motion  or  velocity  is  naught. 

113.  Varignon's  theorem  of  moments. — If  from  any 
point  O,  taken  in  the  plane  of  the  parallelogram  ABCD 

9 


130      COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND  FORCES. 

of  velocities,  and  outside  of  the  angle  BAD,  we  draw  the 
straight  lines  OA,  OD,  and  OC,  the  quadrilateral  OADC 


being  the  sum  of  the  triangles  OAD  and  ODC,  we  shall 
have 

OAC=OAD+ODC-ADC. 

If,  then,  we  let  fall  from  the  point  O  perpendiculars 
Oa  or  Ocj  Ol  and  O<#,  upon  the  sides  AB,  AC,  and  AD, 
we  have  for  the  surfaces  of  the  triangles 


The  above  relation  becomes,  then, 

AC  x  05=  AD  x  Od+  AB  x  Oa. 

The  products  AC  x  O5,  AD  x  Od,  AB  x  Oa  of  the  sides 
AC,  AD,  AB,  by  the  perpendiculars  O5,  Od,  Oa,  let  fall 
from  the  point  O,  upon  their  respective  directions,  are 


COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND  FORCES.      131 

called  the  moments,  and  the  above  relation  shows  that  if 
we  apply  the  preceding  remarks  to  the  component  and 
resultant  motions  of  the  point  A,  we  may  enunciate  the 
theorem  in  saying,  that  the  moment  of  the  diagonal  or  of 
the  resultant  is  equal  to  the  sum  of  the  moments  of  the 
sides  or  components. 

In  the  preceding  figure,  the  two  motions  or  velocities 
tend  to  turn  the  body  in  the  same  direction  around  the 
point  O,  placed  outside  of  the  angle  BAD. 


If  the  point  O  is  within  this  angle  we  shall  then  have 
OAC=OAD-{-ODC—  ADC, 


then 


ODC^DCxOc, 

2i 

and  therefore 

AC  x  O5=  AD  x  Od—  AB  x  Qa. 


And,  as  in  this  case,  the  body,  in  virtue  of  its  two 
motions,  is  urged  in  opposite  directions  around  the  point 
O,  Yarignon's  theorem  may  be  enunciated  in  general, 
being  thus  extended  for  any  number  of  simultaneous 


132 

motions  or  velocities,  in  saying  that  the  moment  of  the  re- 
sultant is  equal  to  the  sum  of  the  moments  of  the  compo- 
nents, which  tend  to  turn  the  body  in  one  direction,  minus 
the  sum  of  the  moments  of  components  tending  to  turn  it 
in  an  opposite  direction,  or  more  generally,  that  the  mo- 
ment of  the  resultant  is  equal  to  the  sum  of  the  moments 
of  the  components,  provided  that,  taking  as  positive  the 
moments  relative  to  a  certain  direction  of  motion,  we 
agree  to  adopt  as  negative  those  which  belong  to  an  op- 
posite direction. 

114.  Extension  of  these  theorems  to  bodies  or  systems 
impressed  with  a  common  motion  of  translation. — All 
that  has  been  said  in  relation  to  a  material  point  applies 
to  bodies  or  material  systems  impressed  with  a  common 
translation,  since  a  determination  of  the  resultant  motion 
or  velocity  of  one  of  the  points  will  give  us  that  of  the 
others.     For  if  all  the  points  are  impressed  with  one  or 
many  common  velocities  in  given  directions,  the  resultant 
velocity  will  be  the  same  for  all. 

115.  Independence  of  the  simultaneous  action  of  many 
forces  upon  the  same  point. — From  observations  which 
show  that  a  material  point  may  be  impressed  with  many 
simultaneous  and  independent  motions  or  velocities,  it  fol- 
lows quite  naturally  that  the  causes  or  forces  which  produce 
these  motions  or  impart  these  velocities    exert   actions 
independent  of  each  other.     Thus  experience  shows  that 
when  a  body  is  subjected  to  the  action  of  many  forces, 
each  one  of  them  communicates,  in  an  element  of  time  t, 
and  in  its  own  direction,  a  small  velocity  v,  proportional 
to  its  intensity,  which  is  the  same  as  if  it  acted  alone, 
whatever  may  have  been  the  previous  motion  of  the  body. 

116.  Case  of  the  forces  acting  in  the  same  direction. — 
When  all  the  forces  act  in  the  same  direction,  the  veloci- 
ties imparted  by  them  being  in  the  same  direction  are 


COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND  FORCES.      133 

added,  and  the  body  is  impressed  with  a  resultant  velocity 
equal  to  the  sum  of  the  component  velocities. 

Now,  these  forces  being  proportional  to  the  velocities 
with  which  they  impress  the  same  body  in  the  same  time, 
it  follows  also  that  all  the  forces  acting  upon  the  material 
point  in  question,  have  a  resultant  equal  to  their  alge- 
braic sum. 

In  fact,  calling  F,  F',  F"  the  forces  acting  in  the  same 
direction  upon  a  material  point  with  a  mass  M  ;  v,  v',  v", 
the  finite  or  elementary  velocities  imparted  by  them  in 
the  same  time,  we  have 


P_          ™  ™_ 

=~T'       ~T>       : 

and  as  the  resultant  velocity  is 


we  have,  calling  R  the  resultant  of  the  forces, 


or 

K=F+F'+F"+&c. 

Moreover,  if  we  multiply  this  last  relation  by  the  space  s 
described  by  the  material  point  in  the  common  direction 
of  the  forces  and  their  resultant,  we  have 

Ifc=F«+F'«+F"«+&c., 

an  expression  showing  that  the  work  of  the  resultant  is 
equal  to  the  algebraic  sum  of  the  works  of  the  compo- 
nents, acting  either  as  motive  or  resistant  works.  (No.  93.) 
Finally,  in  order  that  the  motion  of  the  body  may  be 
uniform,  it  is  necessary  that  the  sum  of  the  motive  works 


134:      COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND  FORCES. 

should  equal  the  sum  of  the  resistant  works,  which  leads 
to  the  relation 

K=F+F'+F"+&c.=0, 
or 


which  expresses  that  the  result  is  nothing,  or  that  the 
work  of  the  motive  forces  is  equal  to  that  of  the  resisting 
forces. 

Equilibrium  is  but  a  particular  case  of  uniform  motion, 
and  when  the  velocity  is  zero  the  preceding  condition  is 
also  that  of  equilibrium. 

117.  Case  where  the  forces  acting  upon  the  body  have 
different  directions. — We  have  seen  by  the  examples  of 
No.  100,  relative  to  the  fall  of  bodies  impressed  at  the 
same  time  with  a  horizontal  motion,  that  the  velocities 
imparted  in  different  directions  were  wholly  independent 
of  each  other. 

In  obedience  then  to  the  simultaneous  action  of  these 
forces,  the  body  will  receive  the  velocities  A£,  A^,  pro- 
portional to  their  intensities,  and  in  the  direction  of  the 

forces,  and  these  com- 
ponent velocities  will 
have  a  resultant  which 
will  be  the  diagonal 
of  the  parallelogram 
Abed.  If  we  take  AB 
and  AD  proportional 
to  the  velocities  Ab 

J[        £ jo >jp    and  Ad  to  represent 

Fie.  49.  the  forces  P  and  Q 

producing  these  small  velocities,  the  resultant  of  these 
forces  to  which  the  resultant  velocity  is  due,  will  be  pro- 
portional to  the  velocity  imparted  in  the  same  time  and 
in  the  direction  of  its  action,  or  to  Ac ;  we  have,  then, 


COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FORCES.      135 


Then  the  resultant  R  will  be  represented  in  magnitude 
and  direction  by  the  diagonal  AC  of  the  parallelogram 
ABCD. 

Then  the  resultant  of  two  forces  acting  simultaneously 
upon  the  same  body  is  represented  in  magnitude  and  di- 
rection by  the  diagonal  of  Q 
a  parallelogram  construct- 
ed upon  these  two  forces. 
Reciprocally,  every  force 
can  be  resolved  into  two 
others,  in  any  two  arbitrary 
directions,  equal  to  the 
sides  of  the  parallelogram 


FIG.  50. 


whose  diagonal  is  the  given      . 
force,  and  whose  sides  are 
parallel  to  the  given  directions. 

If  the  two  directions  are  perpendicular  to  each  other 
we  have 


=E  cos  CAB,  Q=R=K  cos  CAD. 


118.  Quantity  of  work  of  a  force  whose  point  of 
application  does  not  move  in  the  same  direction  as  the 
force. — When  a  force  R  does 
not  act  in  the  same  direction 
of  a,  the  path  described  by 
its  point  of  application,  it 
can  be  resolved  into  two ; 
the  one  P  represented  by 
AB  in  the  direction  of  this 
path;  the  other  Q,  repre-  Fro- 6L 

sented  by  AD,  perpendicular  to  it.  The  work  of  P  will 
be  P  x  Aa.  Designating  by  Aa  the  path  really  described, 
the  work  Q  will  be  zero,  since  it  has  no  motion  in  its 
own  direction.  Then  the  work  of  the  force  R  will  be 


136      COMPOSITION   OF  MOTIONS,  VELOCITIES,  A3TD   FORCES. 

measured  by  that  of  its  component  P.  But  in  dropping 
the  perpendicular  db  upon  AC,  we  have  by  the  similar 
triangles  ACB  and  Aab. 

K  :  P  :  :  Aa  :  AJ,  whence  ~R.Ab=P.Aa. 

Consequently,  the  work  of  the  force  H  may  be  measured 
by  that  of  its  component  P  in  the  direction  of  the  path 
described,  or  by  the  product  of  its  intensity  K  into  the 
projection  Ab  of  the  path  Aa  upon  its  own  direction. 

119.  Application  of  Varignorfs  theorem  to  forces.  — 
Since  the  resultant  of  two  forces  is  represented  in  magni- 
tude and  direction  by  the  diagonal  of  the  parallelogram 
constructed  upon  these  forces,  as  sides,  it  follows  that  the 
purely  geometrical  theorem  of  Yarignon  applies  to  forces 
as  well  as  lines,  and  that  consequently, 

The  resultant  of  two  or  any  number  of  forces  acting 
in  the  same  plane  has  for  its  moment,  in  relation  to  any 
point  of  this  plane,  the  sum  of  the  moments  of  the  forces 
which  tend  to  turn  it  one  direction,  minus  the  sum  of  the 
forces  tending  to  turn  it  in  the  other  direction. 

"Which  is  expressed  by  the  relation 


—  &c. 
In  calling 

P,  P',  ....  the  forces  tending  to  turn  the  body  in  one 
direction,  and  p,pr  the  respective  lever  arms  of  these 
forces  ; 

Q,  Q',  ....  the  forces  tending  to  turn  the  body  in  the 
other  direction,  and  q,  qf,  ----  the  respective  lever  arms  of 
these  forces  ; 

R  the  resultant  and  r  its  lever  arm. 

120.  The  resultant  work  of  any  number  of  forces  is 
equal  to  the  algebraic  sum  of  its  component  works.  —  In 
the  most  simple  case,  when  the  forces  all  act  in  the  di- 


COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND   FORCES.      137 


rection  of  the  path  described,  the  resultant  of  all  the 
forces  is  evidently  equal  to  the  sum  of  those  acting  in  one 
direction  minus  the  sum  of  those  acting  in  an  opposite 
direction,  and  as  the  path  described  by  their  points  of 
application  is  the  same,  the  proposition  is  evident. 


121.  Forces  acting  in 
any  direction. — If  we 
first  consider  the  forces 
P  and  Q  and.  their  re- 
sultant R  as  respective- 
ly proportional  to  'the 
lengths  AB,  AD,  and  Jr 
AC,  and  AM,  the  direc- 
tion of  the  path  describ- 
ed, and  project  P,  Q 
and  II  upon  this  direc- 
tion, we  shall  have 


AB'=F,  AD'=:Q'  and  AC'=R' 

for  the  components  in  the  direction  of  any  path  described, 
A#,  for  example,   and  the  work  of  these   components, 
which  is  equal  to  that  of  the  primitive  forces  P,  Q  and 
R,  will  be  respectively  P'.Aa,  Q'.A&,  R'.A&. 
Now  it  is  evident,  according  to  Fig.  52,  that 


FIG.  52. 


^R'^AB'  x  B'C'=P'+Q'. 
Thus  in  the  case  of  this  figure, 


In  the  case  of  Fig.  53  we  have 


^R'^AB'—  AD^P'—  Q', 


and  consequently 


138      COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND   FORCES. 


The  difference  of  these  two  results  arises  from  the  fact 
that  in  the  first,  the  two  forces  P  and  Q  act  both  in  the 
direction  of  the  path  described,  while  in  the  second,  the 
force  Q  acts  in  an  opposite  direction  and  occasions  a  re- 
sistant work. 

Further  proof  of  this  result  is  derived  from  the  fact 
that  the  projection  of  the  resultant  is  equal  to  the  alge- 
Draic  sum  of  the  projections  of  the  components  upon  any 
line  in  the  direction  of  the  path  really  described,  and  the 
multiplication  of  the  two  members  of  this  equality  by  the 
space  described,  is  an  expression  of  the  following  general 
theorem. 

When  a  material  point  is  acted  upon  ~by  any  number  of 

forces,  situated  in    the 
*  /  same-  plane,  which  tend  to 

impart  a  motion  of  trans- 
lation, the  work  developed 
~by  the  resultant  is  equal 
to  the  sum  of  the  works 
of  the  forces  which  urge 
the  body  in  the  direction 
of  the  path  described, 
minus  the  sum  of  the  works  of  the  forces  which  urge  it 
in  an  opposite  direction. 

Without  entering  into  theoretic  developments  which 
are  foreign  to  the  special  purpose  of  this  treatise,  we  re- 
mark that  analogous  reasonings  apply  to  the  case  of  many 
forces  acting  upon  the  same  body,  in  any  direction  in 
space. 

The  elementary  work  being  termed  the  virtual  mo- 
ment, the  above  enunciation  may  be  thus  stated,  that  the 
sum  of  the  virtual  moments  of  the  components,  taken  with 
the  proper  sign,  is  equal  to  the  virtual  moment  of  the  re- 
sultant :  which  is  the  principle  Jcnown  as  that  of  virtual 
velocities. 


FIG.  53. 


122.  Case  where  the  point  tends  to  turn  around  a 


COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FORCES.       139 


point  or  a  fixed  axis.  If  the  point  O,  from  which  is  let 
fall  the  perpendicular  upon  the  direction  of  the  two  forces 
P  and  Q,  Fig.  54,  is  the  projection  of  the  axis  of  rotation, 
or  the  point  around  which  the  plane  of  the  forces  and  the 
body  tend  to  turn,  the  relation  of  the  moments  (No.  113), 


K  x  0&=P  x  00  ±Q  x  Oc  or  "Rr= 
making  OJ=r,  Oa=j>  and  Oc=([, 


becomes  by  multiplying  all  the  terms  by  the  arc 
described  at  a  unit  of  distance 


Rra  — 


Now  ral}  pa»  ga^  are  respectively  the  elementary  or 
finite  arcs  described  by  the  foot  of  the  perpendicular  or 

the  paths  described  by  the 
point  of  application  of  the 
forces  K,  P  and  Q,  in  their 
proper  directions,  and  con- 
sequently Kra^  Pj?^,  and 
Q^,  are  the  works  re- 
spectively developed  by 
these  forces,  and  the  above 
relation  demonstrates  for 
motion  of  rotation  the 
proposition  already  established  for  motions  of  translation. 


FIG.  54. 


123.  Conditions  of  uniform  motion  or  equilibrium. 
Case  where  all  the  forces  are  contained  in  the  same  plane. 

If  the  material  point  considered  is  acted  upon  by 
forces  in  ^the  same  plane,  it  must  remain  in  this  plane, 
and  at  any  instant  it  can  only  act  in  obedience  to  a  motion 
of  translation,  or  to  one  of  rotation,  or  to  these  two  com- 
bined. 

Since  every  motion  of  translation  may  be  resolved 
into  two  others  in  the  same  plane,  the  real  motion  of  the 


140      COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND   FORCES. 

material  point  will  be  uniform  if  its  two  components 
are  so. 

Then  the  condition  of  uniform  motion  of  translation  is 
the  same  as  that  of  uniform  motion  in  any  two  given  di- 
rections. This  latter  will  be  fulfilled  if  the  forces  or  their 
components  urging  the  material  point  in  the  direction  of 
said  axis,  while  acting  for  the  acceleration  of  its  motion, 
develop  a  work  equal  to  that  of  the  force  which  retards 
it ;  or,  in  other  words,  the  sum  of  components  acting  in 
one  direction  must  be  equal  to  the  sum  of  those  acting  in 
an  opposite  direction,  or  their  algebraic  sum  must  equal 
zero,  according  to  the  previously  established  condition. 

Then  the  motion  of  translation  of  a  material  point  will 
be  uniform  when  the  respective  sums  of  the  component 
forces  soliciting  it  in  any  two  directions  within  the  plane 
shall  ~be  separately  equal  to  zero. 

Equilibrium  being  but  a  particular  case  of  uniform 
motion  where  the  velocity  is  zero,  the  same  conditions 
exist  for  it  as  for  uniform  motion  of  translation. 

In  rotation,  it  is  evident,  that  if  all  the  accelerating 
forces  in  the  direction  of  the  motion  develop  a  work  equal 
to  that  of  the  retarding  forces  iii  an  opposite  direction,  the 
motion  will  be  uniform,  that  is,  for  uniform  motion  of  ro- 
tation, the  sum  of  the  moments  of  the  forces  tending  to 
produce  motion  in  one  direction  must  be  equal  to  the  sum 
of  the  moments  tending  to  produce  rotation  in  the  opposite 
direction. 

Equilibrium  being  but  a  particular  case  of  uniform 
motion  where  the  velocity  is  zero,  the  same  conditions  exist 
for  the  equilibrium  of  any  number  of  forces  situated  in 
the  same  plane. 

124.  Case  where  the  forces  act  in  any  manner  in 
space. — The  motion  of  bodies  is  generally  composed  of 
one  of  translation  and  one  of  rotation  round  a  certain 
point,  and  since  the  motion  of  translation  must  be  uniform 
so  long  as  the  three  motions  resolved  parallel  to  three 


COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FOKCES.       141 

perpendicular  axes  are  uniform,  we  are  led  to  the  condi- 
tion, that  the  sums  of  the  works  developed  in  the  direction 
of  each  axis  must  separately  be  zero. 

In  respect  to  motion  of  rotation  we  remark,  that 
generally  the  point  around  which  a  body  rotates  varies  at 
each  instant :  on  this  account,  we  give  it  the  name  of  the 
centre  of  instantaneous  rotation.  This  being  the  case,  it 
is  evident  that  the  rotation  around  any  centre  may  be  re- 
solved into  rotations  around  the  three  preceding  axes,  or 
axes  parallel  to  them,  through  the  centre  of  instantaneous 
rotation  at  the  moment  of  its  consideration.  Moreover, 
the  resultant  motion  of  rotation  will  be  uniform,  if  the 
components  are  so.  The  rotation  around  these  axes  being 
due  to  the  components  perpendicular  to  each  axis,  uni- 
formity of  motion  will  take  place  if  the  sum  of  the  mo- 
ments of  the  component  forces,  respectively  parallel  to 
the  two  axes  taken  in  their  relation  to  the  third  are  sepa- 
rately equal  to  zero  :  this  leads  to  the  relation  between 
the  moments  which  must  be  taken  successively  in  their 
relation  to  each  of  these  axes. 

The  general  condition  of  uniform  motion  of  a  material 
point  solicited  by  any  two  exterior  forces  is  then  reduced 
to  the  following : 

1.  The  sum  of  the  component  works  in  the  direction  of 
any  three  rectangular  axes  must  equal  zero. 

2.  The  sum  of  the  moments  of  the  given  forces  in  rela- 
tion to  these  three  axes  must  be  separately  equal  to  zero. 

Equilibrium  being  but  a  particular  case  of  uniform 
motion  where  the  velocity  equals  zero,  these  conditions 
exist  also  for  the  equilibrium  of  forces. 

The  preceding  discussion  shows  that  the  study  of  mo- 
tions produced  by  any  forces  may  always  be  reduced  to 
that  of  translation  in  the  direction  of  the  forces  or  their 
components  and  of  the  motion  of  rotation  around  a  given 
axis.  We  have  already  examined  the  first  of  these  mo- 
tions, and  will  now  investigate  the  second,  but  first  it  is 


142      COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FOECES. 

proposed  to  extend  the  theorem  to  the  case  of  parallel 
forces. 

125.  Parallel  forces— -We  have  seen  (No.  108)  that 
the  projection  of  the  resultant  of  any  number  of  forces 
applied  to  the  same  material  point,  upon  any  right  line, 
is  equal  to  the  algebraic  sum  of  the  projections  of  these 
same  forces  upon  the  same  straight  line.  The  demon- 
stration of  this  proposition  being  entirely  independent  of 
the  direction  of  the  forces,  and  the  angles  contained  be- 
tween them,  and  with  their 
/  Q  ^^  resultant,  it  must  be  true 

A" : -^7?^         also  when  we   make  the 

projection  upon  the  re- 
sultant itself,  whence  it 
- '  follows,  that  the  resultant 
of  any  number  of  forces 
applied  to  the  same  point 

is  equal  to  the  algebraic  sum  of  these  forces  acting  in  its 
own  direction. 

This  may  also  be  shown  from  Fig.  55,  from  which  we 

have 

AC  or  R=CD'+D'A=:AB'+AD', 
or 

R=P'+Q', 

in  calling  P'  the  projection  AB'  of  AB  or  P  upon  AC  and 
Q'  the  projection  AD'  of  AB  or  Q  upon  AC.  The  pro- 
jections P'  and  Q'  of  the  forces  P  and  Q  upon  the  direc- 
tion of  the  resultant  are  moreover  evidently  the  compo- 
nents of  these  forces,  in  the  direction  of  their  resultant.  In 
the  case  where  the  angle  formed 
by  the  direction  of  the  forces  P 
and  Q  is  obtuse,  it  is  easy  to  see 
that  the  proposition  of  No.  180 
holds  good,  and  the  one  just  estab- 
Flo<  56>  lished  is  so  modified  that  the  re- 


COMPOSITION   OP  MOTIONS,  VELOCITIES,  AND  FOECES.      143 

sultant  is  equal  to  the  difference  of  the  projections  of  the 
components.     In  fact,  we  see  by  the  figure  that 

AC  or  K^AB'-CB'^AB'-AD'^P'-Q'. 

126.  Consequences    of  the    composition    of  parallel 
forces.  —  The  preceding  propositions  are  wholly  independ- 
ent of  the  magnitude  of  the  angles  BAG  and  DAC,  or  the 
direction  of  the  forces  P  and  Q  ;  they  hold  good  also  when 
the  point  A  of  meeting  of  the  forces  becomes  more  and 
more  distant,  until  these  forces  becoming  parallel,  it  is 
found  at  an  infinite  distance.    We  have  then  for  two  par- 
allel forces,  in  the  first  case,  when  they  act  in  the  same 
direction, 

E=P+Q, 

and  for  the  second  case,  when  they  act  in  opposite  direc- 
tions, 

K=P-Q. 

127.  Point  of  application  of  the  resultant  of  parallel 
forces.  —  The  theorem  of  moments  (No.  119)  demonstrated 
for  any  point  around  which  forces  tend  to  produce  rota- 
tion, being  also  independent  of  the  direction  of  the  forces, 
must  be  equally  true  for  parallel  forces.     Whence  it  fol- 
lows, that  the  moment  of  the  resultant  of  any  number  of 
parallel  forces,  situated  in  the  same  plane,  in  relation  to 
any  point  in  this  plane,  is  equal  to  the  algebraic  sum  of 
the  moments  of  these  forces,  and  thus  we  are  enabled  to 
determine  the  position  of  the  resultant.     Let  there  be,  for 
example,  two  parallel  forces  acting  in  the  same  directions. 
The  preceding  proposition  becomes 


and  moreover  we  have 


14:4:      COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND   FORCES. 

If  we  take  for  the  centre  of  moments  a  point  of  the 
resultant  itself,  we  have  evidently  r=o  and  Br=o,  and 
consequently  Pp±Q^=o,  which  can  only  be  the  case 
when  we  have  P#>—  Q^,  and  when  the  forces  P  and  Q 
having  the  same  direction  tend  to  produce  rotation  in 
opposite  directions  around  the  centre  of  moments.  Then 
this  point  and  the  resultant  itself  are  comprised  within 
the  directions  of  the  forces  P  and  Q,  and  all  the  perpen- 
diculars to  the  resultant  and  to  the  two  forces  are  divided 
into  parts  reciprocally  proportional  to  these  forces  ;  which 


is  expressed  hv  the  relation  -^=~. 

Q  P 


It  is   the    same  for 


every  secant  drawn  between  the  directions  of  the  forces 
A        CO      JB  ^  an(^  Q  5  wherever  may  be  the  points 
of  application  A  and  B  of  these  forces, 
we  see  that  their  resultant  cuts  this  line 
parts    reciprocally  proportional   to 


n 


a 


their  intensities.  We  have  moreover 
from  the  figure,  in  calling  d  the  dis- 
tance between  the  directions  P  and  Q, 


=d—p  and  Pjp=Q(d—m  p), 


Qd 


whence 

consequently 

whence 


In  the  case  where  the  forces  P  and  Q  are  in  opposite 
directions, 

B=P-Q, 

and  we  have  the  relation 


Now  in  order  that,  in  this  case,  the  forces  P  and  Q, 
which  are  in  opposite  directions,  may  produce  rotation  in 


COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND   FOECES.      145 


"  .....  ~~ 


opposite  directions  in  respect  to  a  point  of  the  resultant, 
the  point  and  the  resultant  itself 
must  be  outside  of  the  two  direc- 
tions of  the  forces.  If  we  call  d 
the  distance  of  these  two  direc- 
tions, we  have  by  the  above  re- 
lations d=q—j}  ;  whence  Q — — 


whence 


or      -?= 


which  gives  the  distance  of  the  resultant  from  the  direc- 
tion of  the  force  P,  and  consequently  its  position. 

If  the  points  of  application  of  the  forces  P  and  Q  are  at 
A  and  B,  the  resultant  cuts  the  line  AB  produced  in  C, 
so  that  these  distances  p  and  q  from  the  directions  of  the 
forces  P  and  Q  are  reciprocally  proportional  to  these 
forces. 


128.  Reciprocally,  every  given  force  may  le  resolved 
into  two  other  parallel  forces  acting  at  given  points.  —  If  a 
force  E  acts  at  a  given  point  C,  of  a  right  line  supposed 
to  be  rigid  and  inflexible,  it  will  always  be  easy  to  find 
the  values  of  two  other  forces  P  and  Q  which  acting  at 
given  points  A  and  B  shall  produce  the  same  effect  as  this 
force.  It  will  be  sufficient  if  we  have  at  the  same  time 

P-t-Q=E  if  they  act,  the  one  at  the  right  and  the  other 
at  the  left  of  E, 

P  —  Q=E  if  the  points  A  and  B  are  both  on  the  same 
side  of  E,  and  in  both  cases 


10 


whence     = 
9 


146      COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FORCES. 

These  two  relations  will  give 


whence  P== 
87  p+q 

d  being  the  distance  of  the  two  given  directions  ;  or 

K,    whence  P=-2-. 
p-q 

Which  indicates  that  the  force  P  acts  in  an  opposite  direc- 
tion to  K. 

The  two  forces  P  and  Q  thus  determined  have  a  single 
resultant  precisely  equal  to  K,  and  may  be  substituted  for 
this  force,  since  they  develop  the  same  work. 

This  decomposition  of  a  single  force  into  two  others  act- 
ing parallel  and  at  given  points  has  frequent  applications 
in  practice.     When,  for  example,  we  wish  to  determine 
the  pressure  that  a  beam  or  a  shaft  of  a  hydraulic  wheel, 
of  a  known  weight,  or  loaded  with  a  given  weight,  exerts 
upon  its  supports,  we  are  led  to  a  resolution  of  this  kind. 

Suppose  a  beam  loaded  at  a 
,    f  ......  t'_  ________     point  C  of  its  length  20,  with  a 

j4  ...........  \&_  _  j^  weight  2P,  and  resting  upon  the 

two  points  of  support  A  and  B, 
situated  at  distances  I'  and  I"  from 
the  point  C.  Calling  P'  and  P" 
the  two  pressures  or  components 

sought,  observing  that  P/+P//=2P,  and  taking  the  mo- 
ments of  the  resultant  and  of  the  components  alternately 
in  relation  to  the  points  of  support  A  and  B,  we  have  in 
the  first  case 

P7' 

P"x2c=2PxZ',    whence   P"=—  , 

o 

in  the  second 

P'x2<j=2PZ",     whence   P'=—  . 

G 

For  a  hydraulic  wheel,  these  two  components  P'  and 


COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND  FOBCES. 


P"  will  be  the  pressures  exerted  by  its  gudgeons  on  its 
bearings  in  virtue  of  the  force  2P. 

Resolving,  then,  all  the  forces  acting  upon  the  axis, 
including  the  weight  of  the  wheel  and  its  shaft,  we  shall 
have,  upon  each  gudgeon,  a  group  of  concurring  forces 
whose  partial  resultant  will  give  the  total  pressure  exerted 
upon  each  bearing. 

129.  Extension  of  the  preceding  theorem  to  any  num- 
ber of  parallel  forces  within  or  not  within  the  sameplane.  — 
The  preceding  theorem  may  be  extended  to  any  number 
of  parallel  forces  by  compounding  the  resultants  of  the 
two  first  with  a  third,  and  so  on  ;  whence  we  conclude  : 

1st.  That  the  resultant  of  any  number  of  parallel  forces 
is  equal  to  the  Bum  of  those  acting  in  one  direction  minus 
the  sum  of  those  acting  in  the  opposite  direction. 

2d.  That  if  the  forces  are  in 
the  same  plane  the  moment  of  j 
the  resultant  in  reference  to  any 
point  in  the  plane  containing  all 
the  forces,  is  equal  to  the  sum  or 
the  difference  of  the  moments  of 
the  components. 

3d.  Also,  if  from  the  points 
of  application  A,  B,  and  D,  of  the 
forces  P,  Q,  and  their  resultant 
R,  we  let  fall  perpendiculars  A  A7, 
BB',  OO',  upon  the  plane,  and  call  Ol  the  point  of  inter- 
section of  the  line  AB  with  this  plane  —  a  point  which 
will  necessarily  be  found  at  the  intersection  of  the  plane 
ABB'  A',  containing  the  perpendiculars  to  the  given 
plane,  it  is  easily  seen  that  from  the  relation  already 
demonstrated 

K.OOI=P.AO1+Q.BOI, 
we  may  deduce 

R.OO'=P.AA'+Q.BB', 


148      COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND  FOKCES. 

that  is,  the  moment  of  the  resultant  in  relation  to  any  one 
plane  is  equal  to  the  sum  or  difference  of  moments  of  the 
components. 

By  considering  the  forces  in  couples,  this  theorem  may 
be  extended  also  to  the  case  of  any  number  of  forces  in 
different  planes. 

130.  The  work  of  the  resultant  of  many  parallel 
forces. — If  the  parallel  forces  are  applied  at  different 
points  of  the  same  body,  and  this  body  is  impressed  solely 
with  a  motion  of  translation,  the  space  described  by  all 
the  points  of  application  of  the  forces  is  the  same,  and  in 
multiplying  all  the  terms  of  the  relation 

K=P+Q-S-T+&c. 


by  the 
A 


o 


space  described,  it  will  be  evident  that  the  work 
of  the  resultant  is  equal  to  the  sum 
of  the  works  of  the  components, 
which  moreover  results  from  the 
theorem  of  No.  120.  If  the  body 
turns  around  a  fixed  axis,  the  same 
proposition  demonstrated  for  any 
forces  is  established  in  a  similar 
manner  for  parallel  forces. 


131.  Centre  of  parallel  forces. — 
The  point  of  application  of  the  re- 
sultant of  any  number  of  parallel  forces,  acting  upon  a 
body,  depends  only  upon  the  ratio  between  their  intensi- 
ties and  not  upon  their  direction.  Consequently,  if  the 
directions  of  forces  change,  their  intensities  remaining  the 
same,  as  well  as  the  respective  direction  of  their  action, 
the  point  of  application  will  remain  in  the  same  position. 
This  point,  which  does  not  change  with  the  direction  of 
the  forces,  is  called  the  centre  of  parallel  forces. 


COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND   FORCES.      149 

132.  Use  of  moments  in  determining  the  position  of 
the  resultant. — The  point  of  application  of  the  resultant  is 
deduced  from  the  relation  of  moments  to  any  point,  right 
line  or  plane.  Calling  r  the  lever  arm  of  the  resultant, 
in  reference  to  a  point,  line  or  plane,  and  K  the  sum  of 
the  moments  of  all  the  forces  in  relation  to  the  same  point, 
line  or  plane,  we  have 

K 


r=- 


T+&C. 


133.  Case  when  all  the  parallel  forces  are  equal  and  in 
the  same  direction.  —  We  have  then 


and 


n  being  the  number  of  forces  equal  to  P. 

It  follows  that  the  lever  arm  r  of  the  resultant  in  rela- 
tion to  any  straight  line  or  plane,  has  for  its  expression 

_Pp+Qq+Ss+&c._P(p+q+s+&c.) 

P+Q+S  +  &C.  =  nP  ' 

or 


that  is,  this  distance  is  equal  to  the  mean  distance  of  aU 
the  points  of  application  from  the  plane  or  given  straight 
line. 

134.  Condition  of  uniform  motion  or  of  equilibrium.  — 
The  motion  of  a  body  solicited  by  parallel  forces  will  be 
uniform  when  the  work  of  the  resultant  is  zero,  which  re- 
quires the  work  of  forces  acting  in  one  direction  to  be 
equal  to  that  of  the  forces  acting  in  an  opposite  direction. 
In  the  case  where  it  tends  to  produce  rotation  around  a 
point  or  an  axis,  we  must  have  in  general 


P.pal-\-Q.qa1  —  S.sa1  —  T.fc^&c.  =0. 


150      COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FOECES. 

If  there  are  only  two  forces  P  and  Q  acting  on  both 
sides  of  the  centre  or  the  axis,  the  condition  of  uniform 
motion  or  equilibrium  is 


or        = 


This  relation  serves  as  the  basis  of  the  theory  of  the 
balance  and  the  lever. 

We  remark  that  the  condition 


gives  Rr=0,  and  may  be  satisfied  whether  T&=o  or  r=o. 
Thus,  that  there  should  be  equilibrium  between  the 
forces  tending  to  turn  a  body  around  an  axis  or  a  point, 
it  is  requisite  and  sufficient  that  the  resultant  be  =o,  (ex- 
cepting the  case  of  couples,)  in  which  case  it  should  pass 
through  the  centre  of  rotation. 

135.  The  balance.  —  This  instrument,  of  such  frequent 
use,  affords  a  simple  application  of  the  principle  of  equi- 
librium. 

The  most  general  disposition  of  balances  consists  of  a 
lever  termed  a  ~beam^  traversed  in  the  middle  of  its  length 
and  perpendicularly  by  a  steel  axle  with  its  extremities  in 
the  form  of  blades  with  rounded  edges,  which  rest  upon 
two  steel  or  agate  plates,  placed  horizontally  upon  the 
supports  of  the  balance.  On  both  sides  of  this  axis  the 
two  arms  of  the  beam  are  equal,  having  at  their  ends  steel 
or  agate  blades,  upon  which  rest  cushions  placed  upon  the 
upper  part  of  the  suspension  of  the  plates,  in  which  are 
placed  the  bodies  to  be  weighed.  Perpendicular  to  the 
beam  and  directly  above  or  below  its  axis,  is  usually 
placed  the  index,  which,  inclining  with  the  beam,  indi- 
cates upon  a  fixed  graduated  limb  the  greater  or  less  in- 
clination of  the  beam.  The  balance  can  only  be  in  equi- 
librium when  the  index  is  vertical,  or  at  zero  of  the  limb. 


COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FORCES.       151 

Notwithstanding  the  simplicity  of  its  arrangement, 
the  balance  is  an  instrument  of  very  difficult  construction, 
and  when  we  are  to  satisfy  as  near  as  possible  the  de- 
mands of  great  precision,  the  conditions  to  be  fulfilled  are 
as  follows : 

1st.  The  balance  must  be  in  equilibrium  when  it  is 
not  loaded,  which  requires  the  two  branches  or  arms  of 
the  beam  to  be  themselves  in  equilibrium  ;  the  length  of 
arms,  measured  from  the  blades  of  the  axle  to  the  blades 
of  the  suspension  of  the  plates,  should  be  the  same,  so  that 
the  equal  weights  may  be  shifted  in  either  plate  without 
inconvenience. 

2d.  The  balance  should  be  sensible  to  a  given  degree, 
according  to  the  use  to  be  made  of  it.  Thus,  delicate  bal- 
ances, designed  for  weights  of  10,  or  20,  and  even  of  40 
pounds,  should  show  at  most  differences  of  .0154  (Troy) 
grains,  and  the  degree  of  precision  should  be  at  least  the 
same  for  all  the  weights,  from  the  smallest  to  the  greatest, 
which  the  balance  is  designed  to  weigh. 

These  conditions  cannot  be  fulfilled,  except  with  great 
care  in  the  construction,  and  the  exact  application  of  cer- 
tain easily  established  principles. 

The  first  requisite  is  that  the  blades  of  the  axis,  and 
the  upper  edges  of  the  blades  of  the  arms  shall  be  parallel 
and  in  the  same  plane,  and  that  the  centre  of  gravity  of 
the  beam  or  of  the  entire  balance,  should  be  always  be- 
low, but  very  near  the  edges  of  the  blades  of  the  axle. 

Let  us  consider  a 
balance  in  which  these  -***- 
conditions  are  not  ful- 
filled, and  suppose  it 
to  be  in  equilibrium. 
Let: 

P  the  weight  of  the  P*P 


body  to  be  weighed  be  Fl€K  61- 

put  in  the  left  hand  plate. 


152      COMPOSITION  OF  MOTIONS,  VELOCITIES,  AND  FOJRCES. 

Q  the  number  of  pounds  and  grains  which  put  in  the 
the  right  hand  plate  establishes  an  equilibrium. 

p  and  q  the  arms  of  the  lever  of  the  weights  P  and  Q, 
at  the  instant  of  equilibrium  ;  we  shall  then  have 

Pp^Q^y  and  if  p=q,  as  it  should  be,  P=Q.     Call 

M  the  weight  of  the  beam,  acting  at  its  centre  of  grav- 
ity G,  supposed  to  be  below  the  axis  of  the  beam. 

m  the  weight  which,  placed  in  the  right  hand  plate, 
inclines  the  beam  and  brings  it  to  the  position  A'OB'. 

In  this  position,  let 

p'  and  q'  be  the  new  arms  of  the  lever,  of  the  forces 
P  and  Q,  which  in  the  case  of  the  figure  satisfy  the  con- 
dition pf>qf. 

g  the  arm  of  the  lever  of  the  weight  M  of  the  beam. 

In  this  new  position  of  equilibrium,  the  condition  of 
equality  of  moments  will  be  expressed  by  the  relation 


make 

We  have  then 


from  which  we  deduce  the  value  of  the  weight  which  in- 
clines the  beam  and  maintains  it  in  this  new  position, 


In  order  that  m  may  be  equal  to  zero,  we  must  have 
p^—o,  or  p'=q'i  and  g—o. 

The  first  of  these  conditions  can  only  be  satisfied  when 
the  two  blades  are  in  the  same  plane,  since  we  see  by  the 
figure  that  then  we  always  have 

p—q  and  p'=qf 


COMPOSITION   OF  MOTIONS,  VELOCITIES,    AND   FORCES.      153 

in  all  positions,  which  shows  the  necessity  of  conforming 
to  this  rule  of  construction. 

ivlSJ 

We  cannot  make  —¥=o,  unless  we  make  g=o,  so  that 

the  centre  of  gravity  of  the  beam  will  be  found  upon  the 
edge  of  the  blades ;  but  then  the  beam  and  scales  will  be  in 
equilibrium  in  all  inclinations,  and  for  all  cases  where 
P=Q.  The  balance  would  be  indifferent,  and  would 
have  no  marked  or  determinate  position  of  equilibrium. 

We  then  relinquish  the  attempt  to  satisfy  wholly  this 
condition  ;  but,  in  order  that  the  balance  may  incline  un- 
der the  smallest  additional  weight  m,  when  P=Q,  we 
make  the  distance  of  the  centre  of  gravity  from  the  axis 
so  small,  that  under  a  given  inclination  of  the  limb,  of 
half  a  degree,  for  example,  the  arm  of  the  lever  g  of  the 
weight  of  the  beam,  shall  be  such  as  that  the  weight  m 
producing  this  inclination  may  be  .0154  or  .0077  grains 
Troy. 

This  result  is  independent  of  the  magnitude  of  the 
weights  P  and  Q,  so  that  the  balance  appreciates  the  ad- 
dition of  .007  grains,  for  all  weights  from  the  smallest  to 
the  greatest,  designed  to  be  measured.  Artisans  have 
attained  this  perfection  by  dint  of  great  care.  M.  Fortin 
has  made  balances  for  weighing  from  2.2  pounds  to  near 
0.015  grains.  The  Conservatory  of  Arts  and  Manufac- 
tures possesses  a  valuable  collection  of  balances,  among 
which  are  two,  one  Fortin's,  the  other  Gambey's,  which 
gives  the  weight  of  from  2.2  pounds  to  nearly  0.015 
grains,  and  one  of  M.  Deleuil,  which  can  weigh  from  22 
pounds  to  nearly  0.007  grains,  and  one  presented  by  the 
United  States  government  which  can  weigh  from  110 
pounds  to  .007  grains  nearly. 

For  assaying  balances,  such  precision  has  been  attained 
as  to  appreciate  a  weight  of  0.0007  grains. 

As  for  balances  used  in  trade,  we  do  not  require  such 
delicacy,  and  well  made  balances,  for  weights  of  110 


154:      COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FORCES. 

pounds,  for  example,  appreciate  a  difference  of  0.15  grains 
only. 

The  general  conditions  of  stability  of  equilibrium  show, 
moreover,  that  if  the  centre  of  gravity  is  above  the  edge 
of  contact  with  the  blades  of  the  axle,  equilibrium  cannot 
subsist,  and  the  balance  will  be  useless. 

The  great  sensibility  of  balances,  due  to  the  conditions 
just  indicated,  as  well  as  to  finish  in  execution,  and  to  the 
polish  of  surfaces,  has  this  inconvenience,  that  their  oscil- 
lations are  slow  and  take  considerable  time  to  arrive  at 
the  position  of  equilibrium.  This  defect  is  remedied  by 
various  contrivances,  having  for  their  object  the  stopping 
of  the  scales  before  being  left  to  the  action  of  the  weights, 
the  placing  of  them  gently  upon  the  edges,  the  stopping 
their  oscillations  at  will,  and  finally  the  limitation  of  their 
amplitude  and  duration. 

For  the  preservation  of  the  form  of  the  knife  edges  and 
their  cushions,  we  should  protect  good  balances  by  some 
disposition  which  enables  us  to  raise  up  the  beam  and  the 
plate  while  we  are  loading  the  scales,  or  when  they  are 
not  in  use. 

136.  Proof  of  balances. — Being  assured  that  the  knife 
edges  are  well  made,  and  contained  in  the  same  plane, 
that  their  cushions  are  well  planed  and  polished,  we  put 
the  beam  in  place,  and  prove  whether  it  is  in  equilibrium 
when  the  index  is  at  zero  or  is  vertical.  We  turn  the 
beam,  end  for  end,  to  see  if  it  is  the  same  in  both  direc- 
tions ;  we  test  the  equality  of  the  arms  of  the  beam,  by 
hanging  the  scales  upon  the  beam,  and  making  sure  that 
the  latter  retains  its  position  of  equilibrium  when  we 
change  their  sides.  "We  load  the  scales  with  weights 
graduated  from  the  smallest  to  the  greatest,  which  the 
balance  is  designed  to  weigh,  and  see  if  its  sensibility  re- 
mains the  same  throughout.  After  one  weighing,  we 
change  the  weights  of  the  plates,  and  see  if  the  results  are 
the  same. 


COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND  FORCES.      155 

The  sensibility  of  balances  for  the'  trades  is  fixed  by 
the  statutes  at  ^oVo-  of  the  weights  to  be  tried,  from  the 
smallest  up  to  the  greatest.  When  the  addition  of  this 
fractional  weight  does  not  incline  the  beam  to  the  side  on 
which  it  is  placed,  the  balance  is  imperfect. 

137.  Method  of  double  weighing.  —  Notwithstanding  all 
the  pains  taken  in  their  construction,  when  it  is  desired  to 
operate  very  exactly,  to  be  free  from  all  liability  to  error, 
we  use  a  very  simple  method  of  the  illustrious  Borda, 
called  double  weighings. 

After  placing  the  body  to  be  weighed  in  one  scale  we 
bring  the  balance  in  equilibrium  by  loading  the  other 
with  weights,  or  scraps  of  lead,  iron,  &c.  When  the  equi- 
librium is  well  established,  we  take  away  the  body  to  be 
weighed,  and  substitute  for  it  a  number  of  units  of  weight, 
required  to  re-establish  equilibrium  ;  we  obtain  the  weight 
exactly,  wholly  disregarding  the  inaccuracies  of  the  bal- 
ance. 

138.  The  steelyard.  —  In  this  system  of  balances  (well 
known  to  the  ancients,  and  of  which  many  light  and  sim- 
ple Chinese  models  are  to  be  found  in  the  Conservatory) 
the  arms  of  the  lever  of  the  load  P,  and  of  the  weight  Q 
are  unequal,^?,  that  of  the  load  remaining  constant,  while 
q  that  of  the  weight  Q  varies;  but  the  counterpoise  is 
always  the  same.    The  condition  of  equilibrium 


is  then  satisfied,  in  varying  the  arm  of  the  lever  q  of  the 
constant  weight  Q,  so  that  we  always  have 


The  ratio       being  constant,  the  arm  of  the  lever  of 


156      COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND  FORCES. 


the  constant  weight,  must  vary  proportionally  with  the 
weight  of  the  body  to  be  weighed.     But  in  the  graduation 


FIG.  62. 


of  the  long  arm  of  the  lever,  we  must  take  account  of  the 
weight  of  the  lever,  and  of  the  scale  ;  which  is  done  in 
determining  first  the  position  of  the  sliding  weight  Q, 
when  the  beam  is  horizontal  or  in  equilibrium,  under  its 
own  weight,  and  that  of  the  hooks  or  scales.  Let  q'  be 
.the  distance  of  this  weight  from  the  axis  ;  the  moment 
Qq'  will  be  equal  to  the  excess  of  that  of  the  empty  scale, 
and  its  arm  of  lever,  above  that  of  the  other  arm. 

If  to  make  equilibrium  with  the  weight  P  in  the  scale, 
whose  moment  is  P£>,  it  is  necessary  to  put  the  running 
weight  Q,  at  a  distance  ^,  we  shall  have  the  relation 


~Pp+Qq'=Qq,  whence  p= 


or 


Thus  in  taking  account  of  the  weight  of  the  apparatus, 
we  see  that  the  distances  of  the  running  weight  from  zero 
increases  proportionally  with  the  weights  to  be  weighed. 

The  lengths  p  and  q',  as  well  as  the  running  weight  Q, 
being  known  when  the  balance  is  made,  we  may  calculate 


COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FORCES.       157 

the  value  of  the  distance  q,  corresponding  to  the  greatest 
weight  to  be  determined  by  the  formula 


but  it  would  be  better,  after  having  thus  calculated  for 
the  maximum  weight  Q',  to  determine  it  exactly  by  ex- 
periment. This  done,  we  divide  the  interval  q—  q'  into  as 
many  equal  parts  as  we  wish  to  have  subdivisions  of  the 
weight  Q'. 

Steelyards  for  weighing  from  40  to  50  pounds  usually 
have  divisions  corresponding  to  the  pound,  and  the  frac- 
tions are  read  at  sight,  according  to  the  position  of  the 
running  weight. 

It  is  moreover  evident,  that  in  this  balance,  as  well  as 
the  preceding,  the  blades  of  the  axis  of  suspension,  those 
of  the  hook  bearing  the  weight,  and  those  of  the  running 
weight,  as  well  as  the  notches  in  which  it  is  arrested, 
should  be  in  the  same  plane,  in  order  that  the  ratio  of  the 
levers  may  be  independent  of  the  inclination  of  the  beam  ; 
the  centre  of  gravity  of  the  latter  should  also  be  a  little 
below  the  axis  of  suspension.  "With  these  conditions  the 
balance  oscillates  freely. 

The  use  of  non-oscillating  (folles)  steelyards  is  forbid 
The  degree  of  exactitude  of  steelyards  is  fixed  by  statute 
a*  5^0-  °f  the  weight  to  be  determined  from  the  smallest 
to  the  greatest. 

139.  Steelyard  with  a  fixed  weight.  (Peson.)  —  "We 
sometimes  nse  for  the  determination  of  small  weights,  a 
balance  having  a  single  plate  and  a  fixed  weight,  arranged 
as  follows  :  a  beam  with  two  unequal  arms,  resting  upon 
a  cylindrical  steel  axle,  receives  upon  its  longest  arm 
the  scale  ;  the  shortest  arm  is  terminated  by  a  fixed  weight, 
usually  of  a  lens-like  form.  An  index  OGr,  fixed  perpen- 
dicularly to  the  length  of  the  lever,  passes  through  the 


158      COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FORCES. 


FIG.  63. 


axis  of  rotation,  bearing  at  its  end  a  weight  q.  The 
different  weights,  the  lever,  the  scale,  the  index,  are  so 

proportioned,  that  when  the 
plate  is  empty  and  the  beam 
horizontal,  the  centre  of  grav- 
ity of  the  apparatus,  composed 
of  the  beam,  the  counterpoise 
and  the  index,  is  found  at  a 
point  G  of  the  index,  upon  the 
vertical  passing  through  the 
axis.  In  this  position  the  point 
of  the  index  corresponds  with 
zero  of  the  graduated  limb  of 

the  instrument.  A  weight  P  put  in  the  scale  inclines  the 
lever,  and  we  are  to  determine  the  position  which  the  end 
of  the  index  shall  take  upon  the  limb.  Call  : 

Q  the  total  weight  of  the  lever,  the  counterpoise  and 
the  scale,  &c.,  and  consider  it  as  acting  at  the  centre  of 
gravity  G  of  the  system,  (see  138,)  which  will  then  have 
taken  the  position  G'. 

The  condition  of  equilibrium  of  the  index  will  give  us 

P.B'C^Q.G'H,  whence  P=Q.52*; 

_D  L> 

now  the  triangles  OB'C  and  OG'H  are  similar,  and  we 
have 

OB'  or  OB  :  B'C  :  :  OG'  or  OG  :  OH, 
whence 

fc^-~          RT     OHxOB 

-OG—  ' 
and  consequently 

-TJ        .~      OG     GH       r.     OG 


The  weight  Q  of  the  beam  and  pieces  connected  with 
it  is  known  and  constant,  the  invariable  distance  of  the 
centre  of  gravity  may  be  determined  by  experiment,  the 


COMPOSITION    OF   MOTIONS,  VELOCITIES,  AND   FORCES.       159 

length  of  OB  the  long  arm  of  the  lever  is  known  and  in- 


variable  ;  the  constant  factor  Q  .  —  -  is  then  determined, 

O-t> 

and  we  see  that  the  weights  P  are  proportional  to  the  tan- 
gents of  the  inclination  of  the  lever,  or  to  the  arcs  de- 
scribed by  the  end  of  the  index. 

The  division  of  the  limb  is  then  easy,  since  we  shall 

have 

O~R   P 

tangGOG'=g|.| 

and  making  successively  P=llb>,  l.olb%  2lb%  &c.,  we  may 
calculate  the  values  of  tangents  of  the  angles  GOG7,  an- 
swering to  the  positions  of  equilibrium,  and  consequently 
the  angles  described  by  the  index. 

In  practice,  we  determine  these  angles  by  experiment  ; 
we  see,  however,  that  this  kind  of  balance,  though  not 
susceptible  of  the  same  precision  as  the  common  balance, 
is  yet  quite  handy  for  certain  purposes. 

140.  Quintenz's  Platform  Balance.  —  This  contri- 
vance, bearing  the  name  of  its  inventor,  serves,  accord- 
ing to  its  proportions,  for  weighing  common  bales,  or  the 


FIG.  64 


heaviest  loads.     Its  form  varies  with  its  destination,  but 
its  general  arrangement  is  nearly  always  the  same. 


160       COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FOECES. 

It  is  composed  of  a  horizontal  platform  AB,  resting  at 
one  end  upon  a  triangular  blade  C,  placed  on  the  lever 
DI,  which  is  supported  at  D,  upon  a  fixed  point,  and  is 
sustained  at  I  by  a  vertical  rod  HL.  The  other  end  of 
the  platform  AB  is  sustained  at  F,  by  a  rod  FG,  by  means 
of  the  strut  EF.  The  two  rods  HL  and  FG  are  sustained 
by  blades,  forming  a  part  of  the  lever  HOK,  resting  at  O, 
upon  the  support  of  the  balance,  and  upholding  at  K  the 
platform  on  which  the  weights  are  put. 

This  apparatus  is  usually  borne  upon  a  movable  frame, 
but  is  established  solid  upon  masonry,  when  we  make  use 
of  weighing  bridges,  designed  for  heavy  wagons.  In  all 
cases  the  frame  and  platform  should  be  level  in  all  direc- 
tions. 

The  first  condition  to  be  fulfilled,  is  to  maintain  a 
horizontal  position  when  the  platform  is  loaded,  which  is 
attained  by  giving  proper  proportions  to  the  different 
arms  of  levers.  Indeed,  if  the  point  C  of  the  platform,  or 
the  blade  sustaining  it  is  lowered  the  height  A,  the  point  I 
at  the  end  of  the  lever  AI,  and  consequently  the  point  H, 
the  upper  end  of  the  rod  III,  will  descend  the  height 

k  x  =r-~, ;  but,  at  the  same  time,  the  point  G  of  the  lever 
DO 

OH,  and  so  the  end  F  of  the  rod  GF,  which  sustains  the 
other  end  of  the  platform,  will  be  lowered  the  height 

DI      OG 
XDC?XOH' 
That  the  two  supports  of  the  platform  C  and  F  may 

descend  the  same  quantity,  the  factor  — — -  x  ^-^  must  be 

DO      OH 

equal  to  unity,  which  leads  to  the  condition,  that 

DC7      ,  OG   ,    „  , 

-=-=-  and  ™.  shall  be  equal. 

This  condition  being  satisfied,  we  see  that  in  whatever 
part  of  the  platform  the  load  is  placed,  its  action  upon  the 


COMPOSITION  OF  MOTIONS,  VELOCITIES,  AND  FORCES.      161 

lever  OH  will  be  the  same  as  if  it  were  suspended  by  the 
rod  GF,  upon  the  blade  G.  In  fact,  suppose  the  centre 
of  gravity  of  the  load  rests  upon  O',  if  we  resolve  it  into 
two,  the  one  acting  at  C,  and  the  other  at  F,  and  call  L 
the  vertical  distance  between  C  and  F  : 

P.# 

The  component  of  P  at  F  will  be  -^—  ,  and   will    act 

_L 

directly  upon  G. 

Pv 

The  component  of  P  at  C  will  be  -^,  and  this  will  act 

JL 

at  I,  and  consequently  on  H  with  an  effort  —-x--^-^, 

.L       DL 

«,  ,  .    Py  DC'    HO 

which  occasions  an  effort  at  G  equal  to  --  .—  —  x-^-  ; 

L      Ul      (J(JT 

DC'    HO  ,. 

now  —  -x-=lj  according  to  the  preceding  remarks. 


Then  the  two  components  of  the  load  P  occasion  an  effort 
atG 


since  a?+y=L,  according  to  the  figure. 

Thus,  in  whatever  part  of  the  platform  the  load  is 
placed,  it  is  found  in  virtue  of  the  connections  of  the  sys- 
tem, referred  to  the  point  G,  with  its  integral  value,  and 
we  have  for  the  equilibrium  between  the  load  P  and  the 
weight  Q  the  relation 

P.OG=QxOK. 

Of^ 

The  ratio  -^   is  usually  equal  to  TV  for  the  common 
UJY 

portable  balances  of  commerce,  and  to  T~  for  those  de- 
signed for  the  weighing  of  heavy  loads. 

All  the  fulcrums  and  joints  of  the  system  are  formed 

of  steel  blades,  resting  upon   well  prepared  seats.    The 

blades  corresponding  to  the  larger  parts  of  the  platform 

are  double,  and  exactly  parallel  to  each  other.    Above 

11 


162      COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND   FOECES. 

the  platform  with  the  weight  is  a  small  basin  1ST,  in  which 
we  place  weights  or  scraps,  to  establish  the  equilibrium 
of  the  platform  when  unloaded,  and  of  the  levers  of  the 
apparatus,  when  by  use  or  accidental  causes,  their  primi- 
tive condition  has  been  changed. 

To  save  calculation,  the  weights  used  are  generally 
marked  in  figures,  indicating  the  decuple  or  centuple  of 
their  real  weights,  according  to  the  proportions  of  the 
balance. 

A  stop-lever  with  a  handle  serves  to  raise  the  lever 
HOK,  to  relieve  the  blades,  when  the  balance  is  not 
loaded,  and  to  avoid  shocks  upon  their  edges  at  the  time 
of  loading. 

This  system  of  balance  has  been  modified  in  form,  but 
not  in  principle  ;  it  has  been  ingeniously  attached  to 
cranes,  which  weigh  the  goods  while  being  raised  for 
loading.  Yarious  arrangements  have  been  applied  which 
have  furnished  some  improvements,  and  have  even  regis- 
tered the  weighings,  but  we  cannot  give  a  detailed  account 
of  them. 

141.  Theory  of  the  Lever.  —  From  what  has  been  said 
in  respect  to  a  body  urged  by  parallel  or  concurrent 
forces,  around  a  point  or  fixed  axis,  it  follows,  that  the 
moment  of  the  resultant  is  equal  to  the  sum  of  the  mo- 
ments of  the  components  of  the  forces,  producing  rotation 
in  the  same  direction,  or  to  their  difference,  when  acting 
in  opposite  directions.  The  perpendiculars  r,  p,  q,  being 
let  fall  from  the  centre  of  rotation,  upon  the  direction  of 
the  forces,  are  called  the  arms  of  lever  of  the  forces,  and 
we  have  the  relation 


In  case  of  equilibrium,  we  have  Tt,r=o,  and  conse- 
quently Pj}=Qq,  if  one  of  the  forces  P,  is  the  power,  and 
the  other,  Q,  a  resistance  to  be  overcome  ;  we  see  then, 


COMPOSITION  OF  MOTIONS,  VELOCITIES,  AND  FORCES.       163 

that,  for  equilibrium,  the  moment  of  the  power  must  be 
equal  to  that  of  the  resistance. 

If  the  resistance  and  its  moment  are  given,  the  effort 
to  be  developed  by  the  power  which  is  given  by  the 
formula 


will  be  so  much  the  smaller,  as  the  arm  of  the  lever  p  is 
greater.  This  relation 
contains  the  theory  of 
the  simple  contrivance 
of  the  lever,  employ- 
ed for  the  moving  of 

heavy  weights  by  the  <£f  JP' 

muscular  force  of  men.  Fm- 66- 

We  recognize  three  kinds  of  levers :  that  of  the  first, 
(Fig.  66),  where  the  power  and  resistance  act  on  opposite 
sides  of  the  fulcrum ; 
in  that  of  the  second 
the  power  acts  at  the 
end  of  the  lever,  and 
the  resistance  be- 
tween it  (Fig.  67), 
and  the  fulcrum  ;  in 
that  of  the  third  kind 
(Fig.  68),  the  power 
acts  between  the  ful- 
crum and  the  resist- 
ance;  these  distinctions  have  not  otherwise  any  impor- 
tance, so  far  as  concerns  the  principle  just  enunciated. 

The  advantage  in  the  use 
of  the  lever  is,  that  with  a 
given  and  limited  effort  P, 
we  may  by  a  suitable  pro- 
portion established  between 
the  arms  of  the  lever p  and  p 

q  of  the  power  and  the  re-  FIQ.  es. 


164      COMPOSITION   OF  MOTIONS,  VELOCITIES,  AND  FORCES. 

sistance,  surmount  a  very  considerable  resistance  ;  but  we 
must  bear  in  mind  that  the  arcs  or  spaces  described  by 
the  points  of  application  of  the  forces  being  proportional 
to  the  angles  described,  and  to  their  radii  or  arms  of  lever- 
age, the  relation  of  the  moments  P.^?=Q.^,  multiplied 
by  the  arc  #,  described  at  a  given  unit  of  distance 
f.pa^Q.q^;  which  expression  shows  that  the  work 
of  the  power  is  equal  to  that  of  the  resistance  ;  so  that  in 
the  ratio  of  work  developed  we  have  gained  nothing  by  the 
use  of  the  lever.  It  is  always,  saving  what  is  consumed 
by  the  passive  resistances,  equal  to  that  developed  by  the 
resistance. 

It  is  then  an  error  to  suppose  that  the  effect  of  ma- 
chines, so  far  as  relates  to  their  work,  is  at  all  increased 
by  a  combination  of  levers.  We  must  remember,  thafc  if 
the  efforts  to  be  exerted  diminish  as  the  arms  of  the  lever 
increase,  the  spaces  to  be  described  by  their  points  of  ap- 
plication will  increase  precisely  in  an  inverse  ratio. 


THE  CENTRE  OF  GRAVITY, 

AND   EQUILIBKIUM   OF  TENSIONS   IN   JOINTED   SYSTEMS. 

142.  Application  of  the  preceding  theorems  to  grav- 
ity.— Gravity  acts  upon  all  the  particles  of  bodies,  and 
the  direction  of  all  their  efforts  is  parallel  and  vertical. 
The  resultant  or  sum  of  these  efforts  is  what  is  called  the 
weight  of  the  body.    The  centre  of  parallel  forces,  or  the 
point  through  which  this  resultant  passes,  is  called  the 
centre  of  gravity. 

When  this  point  is  maintained  in  an  invariable  posi- 
tion, so  that  its  resistance  destroys  the  action  of  gravity, 
the  body  is  in  equilibrium  in  respect  to  gravity. 

143.  Determination  of  the  centre  of  gravity. — It  is 
often  necessary  in  the  mechanical  arts  to  know  the  posi- 
tion of  the  centre  of  gravity.     Its  determination  is  made 
by  experiment,  or  by  geometry  and  calculation. 

To  determine  the  centre  of  gravity  of  a  body  by  ex- 
periment, we  place  it  upon  a  sharp  edge,  and  get  by  trials 
the  position  of  equilibrium.  We  mark  upon  the  faces  of 
the  body  traces  of  the  vertical  plane  passing  through  this 
edge,  and  containing  the  centre  of  gravity.  If  needed, 
we  repeat  the  operation  on  many  faces,  and  the  centre  of 
gravity  is  found  at  the  common  intersection  of  these 
planes. 

Sometimes  we  suspend  the  body,  and  with  a  plumb 
determine  the  traces  of  one  or  more  vertical  planes  pass- 
ing through  the  centre  of  gravity. 


166 


CENTEE  OF  GRAVITY. 


This  method  is  attended  with  some  difficulty,  but  still 
is  often  used,  even  for  heavy  weights,  such  as  drawbridges, 
fire-arms,  parts  of  machines  whose  complicated  forms 
do  not  yield  readily  to  the  application  of  geometrical 
methods. 

144.  Geometrical  method. — When  the  bodies  have 
regular  forms,  and  are  homogeneous,  we  may  by  geome- 
try ascertain  the  position  of  the  centre  of  gravity. 

It  is  evident  that  the  centre  of  gravity  of  all  bodies 
with  a  symmetrical  form  is  at  the  centre  of  the  figure : 
thus  the  centre  of  gravity  of  a  plane  sheet,  of  a  cylindri- 
cal or  prismatic  bar,  of  a  sphere,  an  ellipsoid,  or  a  paral- 
lelopiped,  is  contained  in  the  symmetrical  lines  or  planes. 


145.  Triangle. — If  we  suppose  a  triangle  to  be  divided 
into  infinitely  thin  strips,  parallel  to  its  base  BC,  the  cen- 
tre of  gravity  of  each  of  these  strips  being  in  the  middle 

of  its  length,  the  centre 
of  gravity  of  the  triangle 
must  be  found  upon  the 
right  line  AD  passing 
from  the  apex  A  to  the 
middle  D  of  BC,  since  it 
contains  the  centres  of 
gravity  of  all  the  sec- 
tions. For  the  same  rea- 
son,  it  will  be  found  upon 
the  lines  BE  and  OF, 
which  respectively  join  the  summits  B  and  C  with  the 
middle  E  and  F  of  the  opposite  sides.  It  will  be  the  same 
for  the  centre  of  gravity  of  three  equal  balls,  respectively 
placed  at  the  summits  of  the  triangle.  Now  the  centre 
of  gravity  of  the  balls  B  and  C  being  at  D,  and  as  they 
may  be  replaced  at  this  point  by  a  single  weight  equal  to 
their  sum,  the  centre  of  gravity  of  the  system  of  the  ball 


CENTRE   OF   GRAVITY.  167 

A  and  of  the  ball  2B  or  2A,  placed  at  D,  or  in  other 
words,  the  point  of  application  of  the  resultant  of  the 
weights  A  and  2A  will  divide  the  line  AD  into  two  parts, 
reciprocally  proportional  to  2A  and  to  A,  and  will  be 
found  at  two-thirds  of  the  length  AD  from  A,  or  to  a  third 
from  D.  Thus  the  centre  of  gravity  of  a  triangle  is  found 
upon  any  one  of  the  lines,  uniting  one  of  the  summits 
with  the  middle  of  the  opposite  sides,  and  at  a  third  of 
this  line  from  the  base. 

146.  Any  quadrilateral. — The  parallelogram  may  be 
divided  into  two  triangles,  for  each  of  which  the  centre 
of  gravity  is  separately  determined.     "We  then  join  these 
two  centres  by  a  straight  line  divided  into  two  parts,  re- 
ciprocally proportional  to  the  surfaces  of  the  triangles ; 
the  point  of  division  will  be  the  centre  of  gravity. 

147.  Polygons. — We  may  also  determine  gradually 
the  centre  of  gravity  of  any  polygon,  by  means  of  the 
centres  of  gravity  of  the  surfaces  of  triangles  into  which 
it  may  be  decomposed. 

148. — Triangular  pyramid. — In  supposing  the  pyra- 
mid to  be  divided  into  infinitely  thin  sections  and  parallel 
to  one  of  its  bases ;  the  centre  of  grav- 
ity will  be  found  upon  the  straight 
line,  joining  the  opposite  summit,  and 
the  centre  of  gravity  of  the  base, 
which  is  known.  Now  it  willbe  the 
same  with  the  centre  of  gravity  of  //< 
four  equal  balls,  supposed  to  be 
placed  at  the  four  summits  of  the  pyr- 
amid. Compounding  first  the  three  FIG.  TO. 
weights  of  balls,  placed  at  the  summits  of  the  base,  we 
shall  have  a  resultant  proportional  to  the  number  3 ;  it  is 
then  evident  that  the  centre  of  gravity  sought,  or  the 


168  CENTRE   OF   GRAVITY. 

point  of  application  of  the  resultant  of  the  four  equal 
weights,  must  divide  the  line  AC  into  parts  proportional 
to  the  numbers  1  and  3. 

Then  the  centre  of  gravity  of  a  triangular  pyramid  is 
found  in  the  line,  joining  the  centre  of  gravity  of  its  base, 
with  the  opposite  summit,  and  at  a  fourth  of  the  length  of 
the  straight  line  talcenfrom  its  centre. 

The  same  rule  applies  to  a  pyramid  with  any  base. 

149.  Centre  of  gravity  of  a  lody  of  any  form. — We 
often  have  occasion  to  determine  the  centre  of  gravity  of 
a  body  or  volume,  terminated  by  more  or  less  regular 
contours,  but  which  are  not  subject  to  any  known  law. 
Such  is  the  case  with  ships,  for  which  it  is  required  to 
determine  at  once  their  displacement  and  the  centre  of 
gravity  of  the  displacement.  If  we  suppose  the  body  to 
be  divided  into  parallel  sections,  the  moment  of  each  sec- 
tion in  respect  to  the  plane  of  the  first  will  be  given  by 
the  product  of  its  weight  or  volume  into  its  distance  from 
the  plane,  and  the  sum  of  all  the  similar  moments  will  be 
equal  to  the  product  of  the  total  weight  or  volume,  by  the 

_  distance  of  its  centre  of 
/  gravity  from  the  same 
plane.  To  have  the  sum 
\J  of  the  moments,  we  di- 
vide the  length  L  of  the 
body,  in  a  direction  per- 
pendicular to  the  plane,  into  an  even  number  of  equal 
parts,  and  determine  the  area  S  of  each  of  the  sections, 
corresponding  to  the  different  parts  of  the  profile,  and  the 
distance  X  of  the  centres  of  gravity  of  each  of  them  from 
the  plane,  which  will  give  the  products 

S.X.,    8.X.,    S.X.,    &c., 
representing  respectively  the  product  of  the  area  of  each 


CENTRE   OF  GRAVITY. 


169 


of  the  sections,  by  its  distance  from  'the  first  plane,  and 
we  shall  have  for  the  sura  of  the  moments  sought 


+  .  .  .  +  (SX),J 


+2[(SX)3+(SX)5+...(SX)2n_1]]. 

This  sum  should  be  equal  to  the  product  VX,  of  the 
volume  Y  of  the  body  by  the  distance  sought  X  of  its 
centre  of  gravity  from  the  plane  of  the  first  section. 

For  vessels  symmetric  in  relation  to  the  vertical  plane, 
passing  through  the  keel,  it  will  suffice  to  determine  thus 
the  distance  of  the  centre  of  gravity  in  reference  to  two 
planes,  one  containing  the  sternpost,  and  the  other  the 
plane  of  the  water-line,  or  the  lower  plane  of  the  keel. 

150.  The  stability  of  equilibrium.  —  We  have  seen  that 
when  a  body  is  solicited  by  forces  tending  to  turn  it  in 
opposite  directions,  it  is  requisite  for  an  equilibrium  that 
the  resultant  of  these  forces  should  pass  through  the  axis 
or  centre  of  rotation.  But  in  certain  cases  this  condition, 
at  first  satisfied,  will  not  remain  so,  on  the  least  displace- 
ment, and  then  equilibrium  ceasing,  it  is  proper  for  us  to 
examine  in  what  circumstances  the  equilibrium  tends  to 
restore  itself,  or  is  found  to  be  completely  destroyed. 

To  explain  more  readily  this  view  of  equilibrium,  let 
us  consider  an  ovoid  body,  for  ex- 
ample, resting  at  one  of  its  extrem- 
ities upon  a  plane.  "Whenever  the 
vertical  OP,  through  the  centre  of 
gravity  of  the  body,  passes  through 
the  point  of  contact  a  of  the  body 
with  the  plane,  the  point  around 
which  there  is  a  tendency  to  rotate, 
the  body  is  in  equilibrium. 

But  in  the  case  of  Fig.  72,  which  shows  the  major  axis 


1YO  CENTRE   OF   GEAVITY. 

ab  of  the  body  in  the  vertical  position,  with  the  centre  of 
gravity  in  the  highest  possible  position,  we  see  that  by 
the  smallest  displacement,  the  body  will  rest  upon  another 
point  a!  of  its  surface,  nearer  to  its  centre  of  gravity  0, 
and  that  its  centre  of  gravity  having  described  quite  a 
large  arc  of  circle,  will  be  lowered,  and  will  be  so  dis- 
placed that  the  vertical  drawn  through  it  will  not  pass 
through  the  point  of  contact.  It  follows  that  its  weight 
tends  to  turn  the  body  more  and  more,  and  to  withdraw 
it  from  its  primitive  condition  of  equilibrium.  "We  say  in 
this  case,  that  the  equilibrium  is  unstable,  because  as  soon 
as  it  is  broken,  by  any  cause,  the  body  tends  more  and 
more  to  fall.  If,  on  the  other  hand,  the  body  rests  upon 
a  plane  at  a  point  of  its  surface  the  nearest  possible  to 
the  centre  of  gravity,  so  that  its  minor  axis  is  vertical, 
when  we  draw  it  a  little  from  this  position  its  centre  of 
gravity  will  be  raised,  and  its  weight  causing  it  to  re-de- 
scend, the  body  tends  to  resume  its  primitive  position  of 
equilibrium.  In  this  case,  the  equilibrium  is  said  to  be 
stable. 

A  body  is  then  in  the  position  of  stable  equilibrium, 
when,  after  having  been  turned  aside  a  little  from  it,  there 
is  a  tendency  of  itself  to  resume  it,  and  it  is  in  an  unsta- 
ble equilibrium  when,  on  the  contrary,  after  having  been 
a  little  withdrawn,  there  is  a  tendency  to  separate  more 
and  more  from  it. 

Stable  equilibrium  corresponds  generally  to  the  case, 
when  the  centre  of  gravity  is  the  lowest  possible  in  rela- 
tion to  the  form  and  disposition  of  the  body.  Equilibrium 
is  unstable  when  the  centre  of  gravity  can  occupy  a  lower 
position. 

151.  Application  of  the  principles  of  the  composition 
and  resolution  of  forces. — The  foregoing  principles  apply 
particularly  to  constructions.  When  it  is  desired  to  know 
the  pressures,  tensions,  and  thrusts  which  certain  parts  of 
systems  of  constructions  exert  upon  each  other,  either  for 


CENTRE   OF   GRAVITY.  171 

calculating  their  mechanical  effects,  or  determining  their 
dimensions,  so  that  they  may  be  in  a  condition  to  resist 
these  pressures,  and  the  stability  of  the  system  or  con- 
struction may  be  well  established.  We  give  some  exam- 
ples, selected  from  the  most  simple  applications. 

152.  Equilibrium  of  cords. — When  a  cord  or  rod  is 
drawn  at  its  ends  A  and  B,  by  forces  P,  Q . . .  and  P',  Q' . . . 
so  as  to  be  in  equilibrium,  the  / 
components  of  these  forces  per-      ^\Y                ~B/ 
pendicular  to  the  direction  AB,          /~~                ~"\ 
must  have  a  resultant  =  zero,  or   Q/                             H? 
be  in  equilibrium.     As  to   the 

components  in  the  direction  of  the  cord,  the  sum  of  those 
acting  in  one  direction,  from  A  to  B  for  example,  must 
be  equal  to  those  acting  in  an  opposite  direction. 

Thus  the  general  condition  of  equilibrium  of  a  cord  or 
rod,  solicited  by  any  forces,  is  that  all  these  forces  must 
be  reduced  to  two  equal  and  opposite  forces  acting  in  its 
direction.  In  the  case  of  a  cord,  which  offers  no  rigidity 
or  resistance  to  compression,  it  is  further  necessary  that 
they  should  act  in  extension,  and  be  forces  of  tension.  In 
the  case  of  the  rod  or  solid  bar,  this  last  condition  is  not 
necessary,  and  the  efforts  may  produce  extension  or  com- 
pression, but  with  this  condition,  that  the  molecular  reac- 
tion should,  in  the  last  case,  have  sufficient  energy  to 
prevent  changes  in,the  form  of  the  body,  either  from  com- 
pression or  deflection. 

In  either  case,  the  two  equal  and  opposite  forces  so- 
liciting the  rod  give  the  measure  of  tension  it  experiences, 
or  of  the  effort  of  compression  wrhich  it  resists.  These 
rules,  deduced  from  experiment  and  the  theory  of  the 
resistance  of  materials,  show  us  the  proper  proportions  to 
be  given  to  cords  or  to  rods,  in  order  that  they  may  pre- 
sent a  suitable  resistance  to  these  efforts. 

153.  Eqidlibrium  of  the  efforts  transmitted  ~by  cords  or 


172 


CENTRE    OF   GEAVITY. 


rods  meeting  in  the  same  point. — In  this  case,  it  is  requi- 
site and  sufficient  for  equilibrium,  that  any  one  of  the 
efforts  developed  shall  be  equal  and  opposite  to  the  re- 
sultant of  all  the  others.  In  the  case  of  a  support,  it  is 
necessary  that  the  resultant  of  all  the  forces  should  be 
equal  and  opposite  to  the  resistance  opposed  by  the 
support. 

154.  Movable  pulley. — Thus  when  a  street  lamp  is  sus- 
pended upon  the  chape  of  a  movable  pulley  by  a  cord, 
fastened  at  the  fixed  points  A  and  B,  it  acquires  a  position 
of  equilibrium,  and  the  relations  between  the  weight  P 
of  the  lamp,  the  tensions  T  and  T', 
and  the   angles  formed    by  them 
with  the  vertical,  are  determined 
by  the  preceding  conditions. 

If  we  resolve  each  of  the  ten- 
sions T  and  T',  into  two  components, 
the  one  vertical  and  the  other  hori- 
zontal, the  sum  of  the  vertical  com- 
ponents will  be  equal  to  the  weight 
P,  and  the  horizontal  components 
will  be  equal  and  opposite. 

Moreover,  in  case  of  equilibrium, 
exception  being  made  for  friction, 
the  tensions  T  and  T7  will  be  equal ;  which  shows  that 
their  directions  make  equal  angles  with  the  vertical. 


155.  Case  of  a  tower. — If  the  tensions  T  and  T'  are 
those  of  the  chain  of  a  suspension  bridge,  passing  over  the 
roller  0,  through  which  they  act  upon  the  support  AB, 
their  resultant  P  must  act  in  the  direction  of  this  support, 
and  press  upon  its  base ;  for  if  it  passes  without  in  a  line 
CD',  for  example,  it  will  tend  to  turn  it  around  its  lower 
edge  B,  and  to  upset  it.  Though  the  weight  of  the  tower 
itself  is  opposed  to  this  rotation,  it  is  at  least  prudent  and 


CENTRE   OF   GRAVITY. 


173 


even  necessary  to  satisfy  this  condition,  that  the  resultant 
of  the  tensions  shall  fall  within  the  base  of  the  tower,  as 
near  its  centre  of  gravity  as 
possible ;  this  object  is  at- 
tained, either  in  the  con- 
struction of  the  support,  or 
by  giving  proper  directions 
to  the  tensions. 

156.  The  funicular  poly- 
gon.— By  this  name  we  de- 
signate a  polygon  formed  of 
cords,  of  chains,  and  of  rods, 
whose  summits  are  solicited  by  any  forces  whatever.  The 
equilibrium  of  such  a  system  is  subject  to  rules  easily  de- 
termined. 

Suppose,  at  first,  that  the  polygon  has  not  all  its  sides 
in  the  same  plane,  and  is  represented  by  the  contour 


JJS 

FIG.  75. 


FIG.  76. 

ABCDEF.    Let  P,  Q,  E,  S,  T  .  .  .  be  the  forces  acting 
upon  each  of  the  summits  A,  B,  C,  D,  E; 

T,  T1?  T2,  T3,  T< . . .  the  respective  tensions  of  the  sides 
AB,  BC,  CD,  DE,  EF. 

If  the  entire  polygon  is  in  equilibrium,  it  must  be  so 
for  all  the  summits  separately,  and  thence  applying  the 
reasoning  of  Art.  153  we  see,  1st,  that  any  two  sides 
uniting  at  the  same  summit,  and  the  direction  of  the  force 
acting  upon  this  summit,  must  be  in  the  same  plane ;  2d, 
that  the  tension  T^  must  be  equal  and  opposite  to  the  re- 


174  CENTRE   OF   GKAVITY. 

sultant  of  the  force  P  and  of  the  tension  T ;  that  this  same 
tension  T\  must  be  equal  and  opposite  to  the  resultant  of 
the  force  Q,  and  of  the  tension  T2,  and  consequently  in 
substituting  at  the  summit  C  for  the  tension  T^  its  two 
components  P  and  T,  the  forces  P  and  Q  and  the  tensions 
T  and  T2,  supposed  to  be  transferred  to  the  point  C,  par- 
allel to  themselves,  must  be  in  equilibrium.  "We  see  also 
that  the  forces  P,  Q,  K  and  the  tensions  T  and  T3,  sup- 
posed to  be  transferred  to  the  summit  D  in  parallel  direc- 
tions, must  then  produce  an  equilibrium. 

By  continuing  this  process,  we  arrive  at  the  conclusion, 
for  the  equilibrium  of  the  funicular  polygon,  that  when 
all  the  external  forces  and  the  tensions  of  the  extreme 
sides  are  regarded  as  transferred  parallel  to  themselves  to 
any  summit,  they  must  necessarily  produce  an  equilibrium 
there. 

The  preceding  remarks'  are  independent  of  the  direc- 
tion of  the  forces,  and  the  nature  of  the  sides,  and  are 
applicable  when  the  sides  are  subjected  to  efforts  of  com- 
pression instead  of  tension,  with  the  reservation,  solely, 
that  the  sides  be  sufficiently  rigid  to  resist  the  compres- 
sions, without  a  change  of  form. 

157.  Case  where  the  forces  soliciting  the  funicular 
polygon  are  weights. — In  this  case  the  forces  P,  Q,  R,  S  . . . 
are  all  vertical  and  parallel,  and  the  polygon  is  necessa- 
rily in  one  plane ;  for,  the  direction  of  each  force,  and 
those  of  the  two  sides  meeting  at  the  summit,  are  in  the 
same  vertical  plane  ;  and,  as  through  one  side  we  can 
draw  but  one  vertical  plane,  it  follows  that  the  two  verti- 
cal planes,  containing  the  same  side  of  the  polygon,  are 
blended,  and  so  for  the  others. 

Moreover,  since  all  the  external  forces  P,  Q,  R,  S  .  .  . 
and  the  tensions  T  and  Tw  of  the  two  outer  sides  are  in 
equilibrium,  these  tensions  must  also  be  in  equilibrium 
with  the  resultant  of  all  the  parallel  forces. 


EQUILIBRIUM    OF   TENSIONS. 


175 


If  the  polygon  is  acted  upon  by  its  own  weight  only, 
as  in  the  case  of  a  chain,  the  resultant  of  all  the  external 
forces  is  the  weight  of  the  polygon,  and  passes  through 
its  centre  of  gravity,  and  the  directions  of  the  tensions  T 
and  Tn  will,  in  case  of  an  equilibrium,  intersect  in  the 
same  point  of  this  resultant,  or  of  the  vertical  of  the  centre 
of  gravity. 


158.  Determination  of  the  tensions  ly  a  graphical  con- 
struction. —  If,  in  accordance  with  these  views,  we  describe 
upon  the  side  AB,  produced  at  a  chosen  scale,  a  length 
equal  to  the  tension  T,  and  construct  the  parallelogram 
Bd$^,  the  side  B#  will  represent  at  the  same  scale,  the  ten- 
sion T,  of  the  side  BC,  and  the  side  B£#  the  external  force 
P.  Then,  if  we  draw  through  the  point  B,  for  example, 
an  indefinite  horizontal,  upon  which  we  lay  off  the  lengths 
BC',  C'D',  D'E',  ET',  F'G',  proportional  to  the  forces 
P,  Q,  R,  S,  U,  and  from  the  same  point  erect  the  line  BO 
perpendicular  to  AB,  with  a  length  proportioned  to  the 


FIG.  77. 

tension  T  or  to  B5,  the  triangle  BOC'  having  two  sides 
BO  and  BC'  respectively  perpendicular  and  proportional 
to  the  sides  B5  and  B<#,  of  the  triangle  B5^,  must  be  simi- 
lar to  it.  Then  the  third  side  OC'  of  the  first  will  be  pro- 
portional to  the  third  side  Id  or  B#,  or  to  the  tension  T,, 
and  it  will  be  perpendicular  to  the  side  BC  of  the  poly- 


176  CENTKE   OF   GRAVITY. 

gon,  the  following  triangle  will  be  in  the  same  condition, 
in  relation  to  the  side  CD,  to  the  force  Q,  and  to  the  ten- 
sion T2,  which  will  be  proportional  to  the  side  OD',  and 
so  on. 

Then,  if  the  weights  soliciting  the  different  summits  of 
a  funicular  polygon  are  in  equilibrium,  and  we  lay  off 
upon  a  horizontal  line  lengths  respectively  proportional 
to  these  weights,  and  then  from  points  of  this  line  corres- 
ponding to  each  summit,  draw  straight  lines  perpendicu- 
lar to  the  directions  of  the  sides  of  the  polygon,  all  these 
right  lines  will  intersect  at  the  same  point,  and  their 
lengths  will  be  proportional  to  the  tensions  of  the  sidea 
of  the  polygon,  which  will  thus  be  determined. 

159.  Suspension  bridges. — Suspension  bridges  afford 
an  example  of  an  application  of  the  preceding  principles 
to  a  polygon,  at  whose  summits  are  fastened  suspension 
rods,  which  sustain  the  roadway  of  the  bridge  on  which 
we  suppose  to  be  uniformly  laid  a  test  load,  established 
by  the  regulations  of  the  government  at  41  pounds  to  the 
square  foot. 

The  polygon  is  formed  of  iron  chains  with  long  links 
of  round  iron,  or  of  iron  plates,  or  by  cables  of  iron  wire, 
stretched  as  parallel  as  possible,  and  bound  with  united 
iron  wires.  Shackles  receive  the  suspension  rods,  which 
are  themselves  made  of  bars  of  iron,  or  of  bundles  of  iron 
wire. 

It  is  important  to  determine  the  form  of  the  polygon, 
or  the  position  of  its  summits,  and  the  tensions  of  its  sides. 
Suppose,  first,  we  have  the  most  common  case,  that  of 
two  towers,  supporting  the  chain  at  the  same  height, 
whose  span  is  divided  into  an  odd  number  of  equal  parts. 
The  lower  side  of  the  polygon  must  be  horizontal.  If 
there  is  but  one  chain  on  each  side,  the  weight  supported 
by  each  pair  of  suspension  rods  is  equal  to  that  of  one  bay, 
so  that  if  we  call  2P  the  weight  of  a  bay,  or  portion  com- 


EQUILIBRIUM   OF   TENSIONS. 


177 


i 
j 

j—  ar~f 

i             i 
!          ! 

1     1     1 

\ 

i  \  ' 

j) 

V 

e' 
P 

2 

FIG.  78. 


prised  between  the  vertical  planes  of  the  two  pair  of  con- 
secutive suspension  rods,  P  will  be  the  load  of  each  rod. 

Calling  T0  the 
tension  of  the  hor- 
izontal side,  this 
tension  must  for 
each  joint  be  in 
equilibrium  with 
the  vertical  forces 
tending  to  turn  the 
sides  which  meet 
them.  This  consid- 
eration enables  us  to  determine  the  relation  between  the 
heights  of  the  different  summits  and  the  distances  from 
the  lowest  summit  E  or  E'  of  each  branch  of  the  polygon. 

Let  y  be  the  height  of  any  summit  B,  above  the  lower 
sideEE7; 

x  the  distance  of  the  vertical  of  this  summit,  from  the 
point  E. 

We  must  have  for  equilibrium  around  the  joint  B,  the 
relation  of  moments 


x  2Z+P  x  31+  .  .  .  +P  .  nl, 


supposing  the  suspension  rod  BJ  to  be  the  nih  division 
from  the  horizontal  side. 

This  relation,  from  the  known  properties  of  progres- 
sions, is  equivalent  to 


If  applied  to  the  tower,  calling  H  its  height  above  the 
horizontal  side,  we  deduce 


12 


178  CENTRE   OF   GRAVITY. 

which  gives  the  tension  of  the  horizontal  side,  showing  it 
to  be  the  less,  as  the  towers  are  more  elevated. 
The  preceding  relation, 


becomes,  observing  that  os=nl, 


P  P 

whence      = 


which  shows  that  the  curve  passing  through  the  summits 
of  the  polygon  is  a  parabola. 

The  summit  of  the  parabola,  which  must  be  symmet- 
rical to  the  right  and  left  of  the  horizontal  side  EE',  has 

for  its  abscissa  x=  —  ~,  since  it  is  to  the  right  of  the  point 

2i 

E,  and  because  we  reckon  the  abscissa  x  on  the  left  of 
this  point  ;  we  deduce  then  for  the  ordinate  of  the  sum- 
mit of  the  parabola 


p/?_n     _  ^p 

J    2T/\4      2/          8T0' 


which  gives  the  position  of  the  summit  below  the  hori- 
zontal side  of  the  polygon. 

The  use  of  several  chains,  on  each  side  of  the  bridge, 
leads  us  to  arrange  them  so  that  one  of  the  summits  shall 
be  at  the  lowest  point  of  the  polygon.  Considerations 
similar  to  the  preceding  will  enable  us  to  determine  the 
relation  of  the  tensions. 

If  we  call  T0  the  tension  of  the  two  pieces  uniting  at 
the  lowest  point,  and  resolve  it  into  two  forces,  the  one 

T'0  horizontal,  the  other  T"0  vertical,  the  latter  will,  for 

p 
each  piece,  be  equal  to  - ,  and  we  shall  have  for  the  ex- 


OF 


EQUILIBRIUM   OF   TENSIONS. 


179 


pression  of  equilibrium  at  any  summit  B,  between  the 
vertical  efforts  and  the  horizontal  tension  T^,  the  relation 


FIG.  79. 

T0'y=PJ+P  x  2Z+P  x  3Z-f  .  .  .  +P  (n-l)  l+^nl, 
or,  according  to  the  known  properties  of  progressions, 


so  long  as  a?=nl. 

This  relation  shows  that  the  curve  passing  through  the 
summits  of  the  polygon  is  still  a  parabola,  and  that  the 
summit  of  this  curve  is,  for  the  case  in  hand,  the  lowest 
point  of  the  polygon. 

H  being  the  height  of  the  towers,  we  deduce  from  this 


"  2H   5 

and  consequently  the  tension  T0  of  the  lower  sides,  since 
the  curve  gives  the  position  of  all  the  summits,  and  the 
inclination  of  the  sides. 

Knowing  the  ten- 
sion of  the  lower  hor- 
izontal side,  it  is  easy 
to  arrive  at  that  of 
all  the  other  sides, 


0 


M     3      z      1     jf~     2T~~3'     lv    by  means  of  the  theo- 
FIO.  so.  rem  of  Art.  158.     If 

we  raise  upon  a  straight  line  M£T,  a  perpendicular  KO, 


180  CENTRE   OF   GRAVITY. 

representing,  at  a  certain  scale,  the  tension  T=P1 . n^n      • 

2M. 

of  the  horizontal  side,  and  on  each  side  of  the  foot  K  of 
this  perpendicular,  lay  off  the  lengths  Kl,  K2,  K3 . . . 
RT,  K2',  K3' . . .  equal,  at  the  same  scale,  to  the  loads 
P,  2P,  3P  . . .  the  lines  Ol,  O2,  O3  . . .  Ol',  O2',  O3' . . . 
will  be  proportional  to  the  tensions  of  the  sides  DE,  CD, 
BC,  &c.,  D'E',  C'D',  B'C',  &c.,  and  will  represent  them 
by  the  scale. 

Now,  the  figure  shows  that 


Ol  ^T^ 

1,  &c., 


which  gives  us  directly  the  tensions  of  the  different  sides 
of  the  polygon  starting  from  the  lower  side. 

"When  one  of  the  summits  is  at  the  lowest  point  of  the 
polygon,  we  use  the  same  construction,  and  have  the  same 
formulae  to  express  the  tensions  T15  T2,  &c.,  by  means  of 
the  tension  T0  of  the  two  lower  pieces. 

160.  Application. — Suppose,  for  example,  that  we 
have  a  bridge  131.23ft>  long,  with  32  suspension  rods, 
3.937ft>  apart,  except  at  the  sides,  which  are  4.59ftt  from 
the  vertical  supports. 

The  width  of  the  platform  is  to  be  16.4ft-,  and  there 
are  four  chains. 

In  conformity  with  the  usual  constructions  the  plat- 
form weighs  672.21lbs-  per  running  foot.  The  test  load 
being  40.977lbs-  per  square  foot,  this  amounts  to  672.21lbs- 
per  running  foot  of  the  four  chains,  or  in  all  per  chain  and 
running  foot  336.1lbs- ;  and  the  spaces  of  the  suspension 
rods  being  3.937ft-  we  have 

P=336.1  x3.937=1323.4lbs 

The  height  of  the  towers  is  16.404:ft-  above  the  lower 
horizontal  side. 


EQUILIBRIUM  OF  TENSIONS.  181 

*  The  vertical  tension  at  each  point  of  suspension  is 
V=±  weight  =  22054lb%  and  the  horizontal  tension  is 
H=Y  cotang  angle  of  suspension  =  44109lb%  and  the 
whole  at  the  end  is  -}/Vf+Rt  =  ^22054?  +4:4:109*=  49314 
pounds  ;  we  have,  then, 


T,  =  V(44109)a  +  19853  =  44153  lbs- 
T2  =  V  441092  +  33083  =  44233 
T3  =  -v/  441092  +  4631*  =  44352 
T4  =  V  441Q92  +  595?  =  44508 
T5  =  V  441092  +  72772  =  M705 
T6  =  V  4±109*  +  86012  =  44938 
T7  =  V  441092  +  99242  =  45212 
T8  =  V  441092  +  112472  =  45521 
T9  =  V  441092  +  125712  =  45865 
T10  =  V  441092  +  138943  =  46244 
Tn  =  |/441092  +  152182  =  46659 


T13  =  V  441092  +  178642  =  47590 

TI4  =  y  441092  +  19187*  =  48101 

T15  =  V  44109s  +  20511*  =  48644 

T16  =  |/  441092  +  220542  =  49312 


*  Morin  has  made  a  marked  error  throughout  this  calculation,  arising  from  taking 
the  number  of  metres  in  the  span,  instead  of  the  number  of  panncls,  nor  in  the  example 
has  he  conformed  to  the  statement  of  Art.  159,  where,  in  the  case  of  several  chains  being 
used,  it  is  recommended  to  bring  the  summit  to  the  lowest  part  of  the  polygon.  The 
common  formula  for  the  horizontal  tensions  of  the  whole  chain  is : 


H  =  G  cotang  a?  =  — 


and  for  the  tension  at  the  ends  is : 


Where  G  is  the  weight  of  the  loaded  half  of  chain,  &  =  half  span ;  a  =  versed  sine,  or 
height  of  arc ;  a?  =  anglo  of  suspension.  I  have  not  followed  literally  the  steps  of  Morin, 
and  have  used  these  formulae,  as  more  direct  and  less  complicated  than  those  given  by 
hi  na. — Translator. 


GENERAL  COMPOSITION   AND    EQUILIBRIUM 
OF  FORCES  APPLIED  TO  A  SOLID  BODY.* 

161.  Forces  applied  to  solid  ~bodies.  —  If  we  refer  to 
what  has  been  said  in  Arts.  11  to  14,  and  the  following, 
upon  the  constitution  of  bodies,  the  mode  of  action  of 
forces,  and  their  point  of  application,  we  readily  perceive 
that  all  the  propositions  relating  to  work,  and  the  compo- 
sition of  forces,  acting  upon  a  material  point,  and  to  the 
conditions  of  uniform  motion  and  of  equilibrium,  may  be 
extended  to  solid  bodies,  composed  of  molecules  or  mate- 
rial points  so  strongly  united  by  the  molecular  attractive 
forces,  that  their  form  may  be  regarded  as  invariable. 

And  first,  let  us  examine  what  occurs  when  a  body  is 
impressed  with  a  motion  of  translation. 

162.  Motion  of  translation  of  a  l^ody  or  system  of 
~bodies  parallel  to  itself. — The  motion  of  a  body  or  system 
of  bodies,  is  called  parallel  translation,  when  all  its  points 
or  parts  describe  simultaneously  equal  and  parallel  paths, 
whether  in  a  finite  time,  or  one  of  infinitely  small  duration. 

In  the  motion  of  translation,  the  elementary  spaces 
described  by  all  the  points  of  a  body  being  equal,  the 
sum  of  the  elementary  works  of  forces  soliciting  the  body 

*  We  borrow  the  demonstration  of  principles,  recited  in  Art.  161,  from 
the  course  of  M.  Poncelet,  at  Sorbonne,  and  from  the  work  of  M.  Reisal,  en- 
titled Elements  of  Mechanics. 


COMPOSITION    AND   EQUILIBRIUM    OF   FORCES.  183 

in  this  direction,  or  the  total  elementary  work  developed 
upon  the  body,  is  equal  to  the  algebraic  sum  of  the  pro- 
jections of  forces  upon  the  common  direction  of  the  path 
described,  multiplied  by  the  elementary  path. 

In  order  that  the  total  elementary  work  may  be  zero, 
or  that  the  motion  of  the  body  may  not  be  changed,  it  is 
only  requisite  that  the  sum  of  these  projections  of  forces 
upon  the  path  described  shall  be  zero. 

This  sum  is  equal  to  the  resultant  of  all  the  forces 
tending  to  produce  the  translation.  This  resultant  should 
then  be  zero,  in  order  that  the  motion  of  the  body  may 
remain  uniform  in  this  direction,  or  that  the  body  may 
maintain  an  equilibrium  in  this  direction. 

163.  Case  of  variable  motion.  —  "When  external  forces 
produce  a  variation  in  the  motion  of  translation,  the  forces 
of  inertia  are  developed,  and  react  against  them. 

If  we  call  jp  the  weight  of  one  of  the  elementary  masses 
composing  a  body,  the  motive  force  and  the  inertia  cor- 
responding to  an  elementary  change  of  its  velocity  will  be 


and  all  the  similar  forces  will  be  parallel,  and  in  the  direc- 
tion of  the  common  velocity  of  translation.  Their  result- 
ant F  will  be  equal  to  their  sum,  and  we  shall  have 


F= 


g          ~/t~~g't~      *? 


P  and  M  being  respectively  the  total  weight  and  mass. 

As  to  the  point  of  application,  all  the  partial  forces  of 
inertia /',/y,/r/,  are  proportional  to  the  weights  p,p' ->$'-> 
&c.,  of  the  different  parts  of  the  body,  and  the  point  of 
application  of  the^r  resultant  will  be  the  same  as  that  of 
the  total  weight,  or  as  the  centre  of  gravity. 


184:  GENERAL   COMPOSITION   AND  EQUILIBRIUM 

Then,  in  the  motion  of  parallel  translation  the  total 
force  of  inertia  is 


9    *         * 

and  its  point  of  application  is  the  centre  of  gravity  of  the 
lody. 

This  consequence  being  independent  of  the  amount  of 
motion  of  translation,  holds  good  for  finite  motions,  and 
for  any  instant  of  motion  in  a  curved  line. 

But  the  resultant  of  the  forces  of  inertia  developed  in 
the  variation  of  motion,  is  by  virtue  of  the  principle  of 
action  and  of  reaction,  precisely  equal  and  opposite  to 
that  of  the  external  forces  producing  this  variation,  so 
that  F  really  expresses  this  resultant,  and  the  relation  be- 
tween the  external  forces,  and  the  forces  of  inertia  in  the 
motion  of  the  translation  is 


. 

g     t         t 

The  acceleration  -  produced  by  the  force  is  given  by  the 
t 

formula 


which  shows  that  it  is  proportional  to  the  force  F,  and 
inversely  proportional  to  the  mass  of  the  body. 

It  is  for  this  reason  that  we  give  considerable  weight 
to  anvils,  and  place  them  on  large  blocks  of  wood,  to 
diminish  the  shock,  and  to  render  the  velocity  imparted 
by  the  hammer  nearly  imperceptible. 

164.  Quantity  of  motion,  and  vis  viva  of  a  l)ody.  — 
We  see,  also,  that  in  parallel  motion,  the  total  quantity  of 
motion  of  a  l)ody  has  for  its  value 


OF  FORCES   APPLIED   TO   A   SOLID   BODY.  185 

From  this  it  follows  that,  yew  a  body  to  receive  a  motion 
of  parallel  translation,  it  is  only  requisite  that  the  result- 
ant of  the  applied  forces  shall  pass  through  its  centre  of 
gravity :  for,  if  it  passes,  otherwise,  the  body  solicited  on 
one  side  by  this  force,  and  on  the  other,  in  an  opposite 
direction,  by  the  resultant  of  the  forces  of  inertia,  which 
passes  through  the  centre  of  gravity,  must  necessarily 
take  a  motion  of  rotation,  at  the  same  time  that  it  does  a 
motion  of  translation. 

It  is  further  evident,  that  the  total  vis  viva  imparted 
to  a  ~body  in  parallel  motion  is  equal  to 

-Va=MV, 

g 

and  is  equal  to  double  the  quantity  of  work  developed  in 
producing  it. 

165.  The  work  of  gravity  in  jointed  or  compound  sys- 
tems.— The  work  of  all  the  components  being  equal  to 
that  of  the  resultant,  it  follows  that  in  the  motion  of 
bodies,  or  of  heavy  jointed  systems,  we  may  substitute  the 
total  work  of  the  resultant  or  of  the  total  weight,  for  the 
work  of  all  the  partial  weights,  and  as  the  work  of  the 
resultant  is  measured  by  the  product  of  the  total  weight, 
and  of  the  space  described  by  its  point  of  applica- 
tion, it  follows  that,  in  machines  or  systems,  with  pieces 
which  ascend  or  descend  under  the  action  of  weight,  the 
total  work  developed  by  the  weight  is  measured  by  the  pro- 
duct of  the  total  weight,  and  the  height,  that  the  general 
centre  of  gravity  is  raised  or  depressed. 

Then,  also,  the  condition  requisite  for  the  descending 
weights  to  be  in  constant  equilibrium  with  the  ascending, 
or  that  the  work  developed  by  one  shall  be  equal  to  that 
of  the  other,  is  that  the  general  centre  of  gravity  must  re- 
main always  at  the  same  height.  Such  is  the  condition  for 
the  equilibrium  of  weigh-bridges,  balance  machines,  &c. 


186 


GENERAL   COMPOSITION   AND   EQUILIBRIUM 


The  principle  set  forth  in  Art.  31,  on  the  measure  of 
work  developed  by  weight  upon  a  body  describing  any 
curve,  ascending  or  descending,  applies  also  to  any  sys- 
tem of  heavy  material  points,  since,  in  this  case,  the  work 
developed  by  weight,  upon  all  the  points,  is  equal  to  that 
corresponding  to  the  elevation  of  the  centre  of  gravity, 
which  is  in  itself  but  a  heavy  material  point. 

166.  A  system  of  any  forces,  acting  upon  a  solid  body, 
may  always  be  red^lced  to  two  equivalent  forces,  applied  to 
two  of  its  points,  one  of  which  may  le  chosen  at  will. — It 
is  readily  seen  that  the  proposed  forces  may  be  resolved 
into  three  others,  applied  at  any  three  points  within  the 
body.  Let  F  be  one  of  these  forces, 
and  0  its  point  of  application  ;  we 
may  resolve  it  into  three  others 
Fa,  F6,  Fc  in  the  directions  AC, 
BO,  CO,  and  suppose  these  com- 
ponents to  be  transferred  to  the 
points  of  application  A,  B,  and  C. 
These  three  components  will  de- 
velop the  same  work  as  their  resultant.  Operating  in  the 
same  way  upon  all  the  other  forces  acting  on  the  body, 
we  shall  have  at  the  points  A,  B,  and  C,  three  groups  of 
concurrent  forces,  which  each  have  a  single  resultant,  and 
the  work  of  these  three  resultants  Ra,  Kfc,  Kc  will  be  equal 
to  the  sum  of  works  of  all  the  forces  applied  to  the  body. 

Moreover,  the  system  of 
three  forces  Ka,  Rj,  Ec,  may 
be  reduced  to  two  forces, 
equivalent  to  the  proposed, 
one  of  which  may  be  ap- 
plied at  a  chosen  point  A. 
Let  us  conceive,  through  the 
point  A,  and  the  directions 
of  the  forces  Rft  and  Rc,  two 
distinct  planes  to  be  passed, 
Upon  this  line  take  a  point 


FIG.  81. 


intersecting  at  the  line  AD. 


OF    FOKCES    APPLIED   TO    A    SOLID   BODY.  187 

A',  and  draw  AB,  resolving  the  force  R6  into  two  others, 
one  in  the  direction  BA',  which  we  transfer  to  A',  the 
other  in  the  direction  BA,  which  we  transfer  to  A. 

We  do  the  same  for  the  force  Rc,  resolving  it  into  two 
others,  one  in  the  direction  CA',  which  we  transfer  to  A', 
the  other  in  CA,  which  we  transfer  to  A. 

The  three  forces  Ka,  Rfi,  Rc,  may  thus  be  represented 
by  two  groups  of  forces,  acting,  the  one  in  A7  with  a  sin- 
gle resultant  R',  the  others  in  A,  the  point  chosen  at  ran- 
dom, and  having  a  single  resultant  R. 

Finally,  the  system  of  all  the  forces  acting  upon  a 
body  may  be  replaced  by  two  equivalent  forces,  one  of 
which  passes  through  an  arbitrary  point  within  the  body. 

It  is  evident,  that  the  work  of  these  two  forces  will  be 
equal  to  the  total  work  of  all  the  forces  applied  to  the 
body. 

167.  Condition  of  uniformity  of  motion,  or  of  equili- 
brium.— The  motion  of  a  body  will  not  be  disturbed,  or 
modified,  by  the  action  of  these  two  forces,  or  of  those 
which  they  replace,  if  the  quantity  of  work  developed  by 
one  is  equal  and  opposite  to  that  developed  by  the  other ; 
which  requires  these  two  forces  to  be  equal  and  directly 
opposite  to  each  other  in  all  possible  displacements  of  the 
body. 

Such  is  also  the  condition  of  equilibrium,  which  is  but 
a  particular  case  of  uniform  motion. 

Reciprocally,  where,  for  all  possible  displacements  of  a 
body,  the  sum  of  the  works  of  the  forces  soliciting  it  is  zero, 
these  forces  will  not  modify  the  motion  of  the  body,  and 
are  in  equilibrium. 

If  among  all  the  possible  displacements  we  conceive 
an  elementary  displacement  of  the  body,  for  which  the 
point  of  application  A  of  the  force  R  remains  fixed,  we 
may  regard  this  point  as  the  centre  of  rotation,  and  the 
point  of  application  A'  of  the  force  R'  will  describe 


188  COMPOSITION   AND   EQUILIBRIUM   OF  FOKCES 

around  A,  with  the  radius  AA'  of  a  circle,  an  elementary 
path  A V,  perpendicular  to  AA'.  But,  since  by  hypothe- 
sis, the  work  of  the  force  K  is  zero, 
its  point  of  application  not  being 
displaced,  in  order  that  the  sum  of 
the  elementary  works  of  E  and  E' 
may  be  zero,  it  is  necessary  that  the 
work  of  E'  should  be  so;  which 
requires  the  path  AV  to  be  perpendicular  to  the  force 
E',  or  that  the  latter  shall  have  the  direction  of  the  line 
AA'.  The  force  R  must  also  be  directed  in  the  same  line. 
In  order  that  the  sum  of  the  elementary  works  of  the 
two  forces  E  and  E',  in  opposite  directions,  may  be  zero, 
as  supposed,  these  forces  must  be  equal  and  opposite,  and 
therefore  will  not  change  the  state  of  motion  of  the  body, 
and  will  be  in  equilibrium. 

Then  the  motion  remains  uniform,  and  the  body  is  in 
equilibrium. 

It  follows  that  three  forces  which  are  not  in  the  same 
plane^  cannot  be  in  equilibrium  ;  for  the  resultant  of  any 
two  cannot  take  the  same  direction  as  the  third. 


MOTION  OF  KOTATION. 

168.  Work  and  equilibrium  of  forces  in  the  motion  of 
rotation  around  a  fixed  axis. — We  have  seen,  (No.  122) 
when  a  material  point  is  subjected  to  the  action  of  many 
forces,  contained  in  the  same  plane,  and  tending  to  turn 
it  around  an  axis  perpendicular  to  this  plane,  that  the 
work  of  the  resultant  of  these  forces  is  equal  to  the  sum 
of  the  works  of  the  components. 

We  arrive  at  a  similar  result  when  the  forces  have  any 
direction  whatever  in  relation  to  the  axis  of  rotation,  for 
if  we  decompose  each  of  these  forces  into  two  others,  the 
one  in  the  direction  of  the  axis,  the  other  in  the  plane 
perpendicular  to  the  axis,  and  passing  through  the  mate- 
rial point,  it  is  evident  that  the  component  in  the  direc- 
tion of  this  axis  supposed  to  be  fixed,  can  only  produce  a 
motion  of  translation,  destroyed  by  the  support  of  the  axis, 
in  consequence  of  which  its  work  will  be  nought ;  the  only 
work  developed  then  will  be  that  of  the  component  con- 
tained in  the  plane  perpendicular  to  the  axis.  It  will  be 
the  same  not  only  for  all  the  forces  applied  to  one  of  the 
material  points  of  the  body,  but  also  for  those  acting  upon 
the  other  parts.  We  have  then  only  to  consider  the  forces 
comprised  in  the  plane  perpendicular  to  the  axis,  and  as 
the  body  is  supposed  to  be  rigid  and  inflexible,  we  are  at 
liberty,  so  far  as  concerns  rotation,  to  suppose  all  the 


190  MOTION    OF   ROTATION. 

forces  comprised  in  one  and  the  same  plane  perpendicular 
to  this  axis. 

If  we  call 

a  the  elementary  arc  described  by  a  point  of  the  body 
situated  at  a  unit  of  distance  from  the  axis  O,  in  an  ele- 
ment of  time, 

r  the  distance  of  any  point  m  from  this  axis, 

The  arc  described  in  the  same  time  by  the  point  m 
will  be  ra. 

If  the  force  F,  acting  upon  this  point,  is  not  perpendicu- 
lar to  the  radius  Om,  we  may  decompose  it  into  two  others, 
one  in  the  direction  of  this  radius,  which  will  be  destroyed 
by  the  resistance  of  the  solid,  since  it  only  has  a  motion  of 
rotation  around  the  axis,  the  other  F',  perpendicular  to 
the  radius,  will  produce  the  only  work  due  to  the  force  F, 
which  will  have  for  its  expression  Fra. 

It  will  be  the  same  for  all  the  other  forces  F1?  F2, 
acting  at  distances  7\,  r#  from  the  axis,  in  planes  perpen- 
dicular to  it.  Their  respective  work  will  be  due  to  their 
components  F'15  F',,  perpendicular  to  the  radii. 

It  results  from  this  that  the  total  work  of  all  the  forces 
acting  upon  a  body  and  corresponding  with  an  angular 
displacement  a  measured  with  the  unit  of  distance,  will 
have  for  expression 


which  is  expressed  in  saying  that  the 
work  is  equal  to  the  sum  of  the 
moments  of  the  exterior  forces  mul- 
tiplied by  the  elementary  arc  meas- 
uring the  displacement  of  points 
situated  at  the  unit  of  distance.  In 
fact,  it  is  evident  that  the  product 
FV  is  equal  to  the  moment  J?op  of 
the  force  F,  as  may  be  easily  proved 


MOTION   OF  ROTATION.  191 

from  the  figure,  and  we  have 

mF  or  F  :  raF'  or  F'  :  :  om  or  r  :  op, 
whence 


The  elementary  work  of  all  the  external  forces  can  not 
then  be  zero,  for  any  angular  displacement  whatever,  ex- 
cept that  the  sum  of  the  moments  of  the  forces  in  relation 
to  the  axis  of  rotation  shall  be  zero,  which  requires  the 
moment  of  their  resultant  to  be  so  likewise,  or  that  it  shall 
pass  through  the  axis  of  rotation. 

What  we  have  said  in  respect  to  this  isolated  case  ap- 
plies to  every  other  axis  of  rotation,  and  consequently,  in 
order  that  the  forces  applied  to  an  invariable  body  shall 
mutually  produce  equilibrium,  or  no  motion  of  rotation, 
the  above  condition  must  be  satisfied  for  any  axis  of  rota- 
tion whatever. 

169.  General  conditions  of  the  uniformity  of  motion 
or  of  equilibrium  of  a  solid  body,  free  in  space,  and  sub- 
jected to  any  forces.  —  It  is  evident  that  a  solid  body, 
entirely  free,  can  receive  and  take  but  one  of  the  three 
following  motions  : 

A  motion  of  translation  without  rotation,  a  motion  of 
rotation  without  translation,  and  a  simultaneous  motion  of 
translation  and  rotation. 

Every  motion  of  translation  may  be  resolved  into  three 
other  motions  similar  in  relation  to  any  three  rectangular 
axes  drawn  in  space,  and  it  is  evident  that  if  each  of 
these  component  motions  is  separately  zero,  the  resultant 
motion  of  translation  will  be  so  likewise,  since  it  will  be 
represented  by  the  diagonal  of  a  parallelepiped,  whose 
sides  are  zero.  This  condition  is  moreover  necessary  and 
sufficient. 

Now,  in  order  that  these  three  motions  shall  be  zero 
for  each  of  these  axes,  the  sums  of  the  components  parallel 


192  MOTION   OF   ROTATION. 

to  the  axes  should  separately  be  zero,  (No.  124.)  Then, 
if  we  call  X,  Y,  and  Z  the  sums  of  the  components  of  the 
exterior  forces  applied  to  the  invariable  solid  considered, 
these  forces  cannot  impart  a  motion  of  translation  if  we 
have  at  the  same  time 

X=O,    Y=O,    Z=0, 

and  the  motion  will  remain  uniform  or  the  body  be  in 
equilibrium  as  to  translation. 

So  also,  every  motion  of  rotation  of  a  body  or  of  mate- 
rial points  composing  it,  may  be  resolved  into  three  mo- 
tions of  rotation  around  three  rectangular  axes  drawn 
through  any  point.  In  order  that  the  body  shall  receive 
no  motion  of  rotation,  it  is  only  requisite  that  the  rotations 
around  each  of  the  three  axes  shall  be  separately  zero, 
which  requires  the  sums  of  the  moments  of  forces  in  rela- 
tion to  each  of  the  three  axes  to  be  separately  zero,  so 
that  if  we  call  L,  M,  and  N  these  three  sums,  we  must 
have  at  the  same  time 

L=O,    M=O,    N=0. 

When  these  conditions  are  satisfied,  the  work  developed 
in  imparting  a  motion  of  rotation  will  be  zero,  and  it  will 
continue  to  move  uniformly  or  will  rest  in  equilibrium. 

In  order  that  the  body  receive  no  motion  of  transla- 
tion, nor  of  rotation,  or  that  its  motion  be  in  no  wise 
altered,  all  that  is  requisite  is, 

1st.  That  the  sum  of  all  the  components  of  the  forces 
soliciting  the  body,  in  relation  to  any  three  rectangular 
axes,  shall  be  separately  zero.  This  is  expressed  by  the 
relations 

X=0,    Y=0,    Z=0, 

L=O,    M=O,    N=0, 

which  we  call  the  six  equations  of  uniform  motion,  or 


MOTION   OF  KOTATION.  193 

the  equilibrium  of  an  invariable  body,  free  and  solicited 
by  any  forces. 

170.  Centrifugal  force. — Every  one  knows,  that  if  we 
tie  a  stone  or  other  heavy  body  to  a  cord,  impress  it  with 
a  circular  motion  of  which  the  hand  is  the  centre,  the  cord 
will  experience  a  tension,  the  greater  as  the  motion  is 
more  rapid.     From  observation  of  this  fact  came  the  use 
of  the  sling  as  an  implement  of  war  among  the  ancients, 
and  which  is  but  a  boy's  play.     Similar  effects  are  seen  in 
wagons  running  swiftly  in  short  curves,  in  circuses,  when 
the  horses  and  riders  are  naturally  induced  to  lean  towards 
the  centre  of  the  curves  they  describe  to  prevent  being 
overthrown.     The  reader  may  readily  find  other  effects 
from  the  same  cause  :  all  of  them  prove  that  in  curvilinear 
motion  the  bodies  are  subjected  to  a  peculiar  force  tend- 
ing to  drive  them  from  the  centre,  which  force  is  called 
the  Centrifugal  force. 

171.  Measure  of  the  centrifugal  force. — To  understand 
what  takes  place  when  a  material  point  is  submitted  to 
the  action  of  the  centrifugal  force,  let  us  examine  first  how 
this  force  is  developed  in  circular  motions. 

When  a  material  point  or  an  elementary  mass  m  passes 
from  one  element  of  a  curve  which  it  describes  to  another, 
it  tends  by  virtue  of  its  inertia  to  continue  its  motion  in  the 
direction  of  the  prolongation  of  this  element,  or  of  the 
tangent  fid  of  the  curve,  and  is  what  is  termed  flying  off 
at  a  tangent,  as  is  the  case  with  the  sling  at  the  moment 
one  suddenly  lets  go  his  hold  upon  the  cord. 

If  the  mass  m  takes  the  direction  of  the  next  element, 
it  is  then  retained  upon  the  curve,  either  by  the  resistance 
of  the  curve  itself,  upon  which  it  then  exerts  a  pressure, 
or  by  the  tension  which  it  develops  in  the  cord.  This 
pressure  or  tension  is  itself  the  measure  of  the  centrifugal 
force,  in  contradistinction  to  which  it  is  sometimes  called 
the  centripetal  force. 

13 


194:  MOTION   OF   ROTATION. 

This  force  is  in  the  direction  of  the  radius  of  the  curve 
or  of  the  corresponding  circle,  and  if  we  call  Y  the  ve- 
locity with  which  the  mass  in  is  impressed  in  the  direction 
of  ab,  and  take  the  length  T)d  to  represent  it,  it  is  clear 
that  the  velocity  destroyed  by  the  resistance  of  the  ci)rd 
or  the  centripetal  force,  will  be  represented  by  the  side 
de  of  the  parallelogram  Icdf,  whose  side  do  is  parallel  to 
the  radius  ob,  in  the  direction  of  which  this  force  is  ex- 
erted. Now,  an  inspection  of  the  figure  shows  that  the 
angles  dbO  and  ~bdc  are  equal  as  internal  and  external, 
and  the  angles  deb  and  cbO  as  alternate  and  internal,  and 
as  moreover  the  angles  cbO  and  aOl)  being  formed  on 
both  sides  of  the  radius  by  two  equal  and  consecutive  ele- 
ments of  the  circle  or  of  the  polygon  whose  infinite  num- 
ber of  sides  replace  it,  it  follows  that  the  angles  ~bdc  and 
deb  are  equal,  and  the  triangle  T)dc 
is  isosceles.  Then  the  velocity  ~ba 
with  which  the  mass  m  is  moved 
in  the  direction  of  the  following 
elements  It,  is  the  same  as  "that  it 
had  in  the  direction  of  the  preced- 
ing element.  Thus  in  circular 
motion,  the  centrifugal  force  does 
not  alter  the  velocity  of  rotation  : 
which  is  conformable  with  the 
principles  upon  work,  which  we 

have  already  recited,  since  this  force,  in  the  direction  of 
the  radius,  or  normal  to  the  path  described,  produces  no 
work  in  the  direction  of  motion,  so  long  as  there  is  no 
path  described  in  its  own  direction  and  by  its  action. 

This  being  settled,  the  velocity  destroyed  in  the  ele- 
ment of  time  t  by  the  centripetal  force  has,  according  to 
the  figure,  dc  for  its  measure,  and  the  centripetal  and  cen- 
trifugal forces,  which  are  equal  and  directly  opposite,  have 
for  a  common  measure 


t 


MOTION   OF  ROTATION.  195 

Now,  the  triangle  fide  and  Obt  having  equal  angles, 
are  similar  ;  we  have  then 

lOilti-.M:  dc, 

whence 

,      Idxlt    Vs 

-^- 


In  calling  R  the  radius  of  the  circle  described,  and  s 
the  elementary  arc  run  over  in  the  element  of  time  t  ;  and 
as  we  have 

V=?  or  s=Vt, 
t 

it  follows  that 

,  _YxY*__V2£ 

K      "E"' 

and  finally,  that  the  centrifugal  force  has  for  its  measure 

Ya 


if,  moreover,  we  call  V\  the  angular  velocity,  or  that  at 
the  unit  of  distance,  we  have  Y=  V^,  and  the  expression 
for  the  centrifugal  force  becomes 


.  ~ 

JLV 

What  we  have  said  of  the  centrifugal  force  applies  to 
a  material  point  describing  any  curved  line,  since  in  each 
of  its  positions,  an  osculating  circle  may  be  substituted  for 
the  curve  ;  the  only  difference  being  in  the  fact  that  the 
radius  K  of  this  circle  varies  for  each  position  of  the  mov- 
ing body,  while  that  in  the  circle  is  constant. 

172.  Work  developed  ly  the  centrifugal  force.  —  "When 
instead  of  being  retained  by  a  circular  curve  or  at  a  con- 
stant distance  from  the  centre  of  rotation,  the  material 


196  MOTION   OF   KOTATION. 

point  is  removed  farther  from  it,  the  centrifugal  force  will 
cause  it  to  describe  a  certain  path  in  the  direction  of  the 
radius  ;  it  develops  upon  this  body  a  work  easily  appre- 
ciated. 

In  fact,  if  in  an  element  of  time  the  material  point  is 
displaced  in  the  direction  of  the  radius  by  a  certain  ele- 
mentary quantity  Y,  the  corresponding  work  of  the  cen- 
trifugal force  will  be 


and  the  total  work  due  to  this  force  when  the  material 
point  shall  have  passed  from  R"  to  R'  at  a  greater  dis- 
tance from  the  centre,  will  be  given  by  the  sum  of  all  the 
analogous  elementary  works  taken  from  R=R"  to  R=R'. 
Now  we  have  seen  by  the  preceding  examples  that  this 
sum  is  equal  to 


--R//2)=w  (V/2-V"2) 


if  we  call  V'^VJR/  and  V'^VJR",  the  velocities  of  ro- 
tation of  the  point  around  the  centre.  We  have  then  for 
the  work  of  the  centrifugal  force 


We  remark  that  the  second  member  of  this  relation  is 
no  other  than  the  variation  of  the  vis  viva  of  rotation,  ex- 
perienced by  the  material  point  while  partaking  of  this 
motion  in  its  removal  from  the  centre  of  rotation,  what- 
ever may  be  the  curve  or  path  described  in  this  removal. 
This  expression  then  could  be  directly  deduced  from  the 
principle  of  vis  viva. 

In  the  case  just  considered,  the  centrifugal  force  tends 
to  increase  the  absolute  velocity  of  the  body  moved,  and 


MOTION   OF  EOTATION. 


197 


acts  thus  as  a  motive  force  which  is  developed  in  the  rno 
tion  of  rotation. 

When,  on  the  other  hand,  the  body  approaches  the 
centre,  the  centrifugal  force  is  opposed  to  it,  and  acts  as  a 
resistance  in  developing  a  work  having  indeed  the  same 
expression,  but  which  is  resistant,  since  the  path  described 
is  in  a  direction  contrary  to  the  action  of  the  force. 

The  preceding  considerations  will  find  their  application 
in  the  study  of  the  effects  of  certain  hydraulic  receivers. 


173.  Action  of  the  centrifugal  force  upon  wagons. — 
When  a  coach  with  great  speed  turns  upon  a  short  curve, 
the  effects  of  the  centrifugal  force  is  felt  by  the  passengers 
who  are  driven  towards  the  outer  curve  with  an  intensity 
often  dangerous  for  those  placed  on  the  outside,  and  which 
may  even  disturb  the  stability  of  the  coach  itself. 

There  is  often  a  prejudice 
against  the  effects  of  this  force 
upon  railways,  wrhen  it  is  pro- 
posed to  use  curves  of  small 
radius ;  but  it  is  easily  shown 
by  figures,  that  in  this  regard 
the  greatest  velocities  with  the 
common  radii  of  curves  pro- 
duce no  danger. 

In  fact,  calling 

P  the  weight  of  the  car  or  any  carriage, 

h  the  height  of  its  centre  of  gravity  above  the  plane 
of  the  track, 

F^-V/R  the  centrifugal  force, 
y 

%c  the  width  of  the  track. 

It  is  evident  that  when  the  car  passes  around  the  cen- 
tre O  of  the  curve,  and  is  arrested  by  some  obstacle,  such 
as  the  falling  or  rising  of  the  rail,  it  tends  to  upset  out- 
wards, in  turning  around  the  point  a  of  instantaneous  sup- 


198  MOTION    OF   ROTATION. 

port.  This  motion  is  counterbalanced  by  the  weight  P 
of  the  carriage,  and  at  the  moment  when  the  weight  and 
centrifugal  force  are  in  equilibrium  as  to  the  point,  we 
have  between  the  moments  of  the  two  forces  P  and 

F^-V^K  the  relation 
9 


which  shows  that,  with  equal  velocities  and  weights,  the 
stability  of  the  car  will  be  so  much  the  greater,  and  the 
equilibrium  better  secured,  as  the  width  2<?  of  the  track  is 
greater  in  its  ratio  with  the  height  of  the  centre  of 
gravity. 

The  velocity  of  transit  answering  to  this  equilibrium 
upon  common  tracks,  for  which  2c—  4.75ft>  with  cars 
whose  centre  of  gravity  when  loaded  is  3.28ft>  in  height, 
and  with  curves  1312ft-  radius  will  "be  given  by  the 
relation 


^K,  whence  YJl^K^ 


a  velocity  beyond  the  greatest  speed  of  railroads.  This 
shows  that  in  this  regard,  the  centrifugal  force  occasions 
no  danger.  But  we  should  not  forget  that  it  brings  the 
flanges  of  the  outer  wheels  to  bear  against  the  rails,  pro- 
ducing a  cutting  away  which  wears  them  out  and  greatly 
contributes  to  their  running  off  the  track. 

174.  Action  of  the  centrifugal  force,  in  fly-wheels.  — 
For  regulating  the  irregularities  of  machines,  we  make 
use  of  rotating  pieces  of  considerable  weight  and  diameter, 
impressed  with  quite  a  great  velocity,  upon  which  the  mo- 

*Morin  has  47^.5;  it  should  be  5SW.24. 


MOTION   OF   ROTATION.  199 

tion  of  rotation  developes  a  centrifugal'force  of  considerable 
intensity. 

Thus,  for  example,  the  fly-wheel  of  an  iron  rolling  mill, 
established  at  the  iron  works  of  Fourchambault,  weighs 
13232lb%  its  radius  is  9.58f%  the  number  of  turns  it  makes 
is  60  in  1',  or  1  per  second. 

"We  have  thus  V,=6.28ft'  in  1",  and  consequently, 

¥^=6.28x9.58. 

If  we  consider  a  segment  of  the  ring  equal  to  £  of  its 
circumference,  corresponding  to  a  single  arm,  its  weight 
will  be  2205  pounds  ;  and  if  its  connection  with  the  adjoin- 
ing segment  is  broken,  the  arm  will  experience,  in  the 
direction  of  its  length,  a  traction  expressed  by 


x  6.282  x  9.58=25887  pounds, 


32.1817 

which  shows  that  in  fly-wheels  the  centrifugal  force  ac- 
quires a  dangerous  intensity,  and  that  it  is  well  to  give 
great  solidity  to  their  connections.  The  velocity  of  rota- 
tion of  these  machines  should  be  confined  within  certain 
limits.  If,  for  example,  we  were  to  impart  to  the  above 
fly  a  double  velocity,  or  120  turns  in  1',  the  centrifugal 
force  of  the  segment  just  considered  would  be  four-fold, 
or  equal  to  103548  pounds. 

175.  Application  to  the  motion  of  water  contained  in 
a  vase  turning  round  a  vertical  axis. — In  this  case  the 
liquid  molecules  are  simultaneously  subjected  to  the  ver- 
tical action  of  their  own  weight,  and  to  a  centrifugal  force 
developed  horizontally;  in  order  that  they  shall  be  in 
equilibrium  under  the  action  of  these  two  forces,  it  is 
requisite  that  the  resultant  of  these  two  forces  should  be 
normal  to  the  surface  assumed  by  the  fluid  mass,  for  if 


200 


MOTION   OF  ROTATION. 


this  resultant  was  inclined  to  the  surface,  the  molecules 
would  yield  to  its  oblique  action. 

Let  us  consider  a  molecule  m 

with  the  weighty  and  mass -,  sit- 

y 
uated  at  the  distance  mp=E,  from 

the  axis  of  rotation  AC.  In  a 
horizontal  direction  and  perpen- 
dicular to  the  axis,  it  will  be  im- 
pressed with  a  centrifugal  force 
expressed  by 


Let  us  take 


mD—p, 


and  construct  the  parallelogram  mBED,  whose  diagonal 
normal  to  the  surface  assumed  by  the  fluid  intersects 
the  axis  at  i.  The  similar  triangles  mpi  and  wBE 
give  us 


whence 


mB  or 


:  BE  OY  p  :  :  mp  or  E  :pi, 
. g 


Thus  the  distance^',  which  is  called  the  subnormal, 
depends  only  upon  the  constant  number  ^,  and  the  angular 
velocity  supposed  also  to  be  constant.  Consequently  this 
distance  is  constant,  which,  according  to  the  known  prop- 
erties of  the  parabola,  shows  that  the  generating  curve  of 
the  surface  of  the  level  is  a  parabola  whose  summit  is  at 
the  point  O,  and  whose  axis  is  that  of  the  rotation,  and 


MOTION  OP  ROTATION. 


201 


we  readily  see  that  its  parameter  is  ~-^  so  long  as  we 
have 


pp'  or  2x  :  mp  or  y  :  :  mp  or  y  :  pi  or  =^-a. 


whence 


1Y6.  Surface  of  water  contained  in  a  bucket  of  a  hy- 
draulic wheel  with  a  horizontal  axle. — In  following  the 
reasoning  of  the  preceding  case,  it  is  easy  to  see  that,  if  we 

represent  by  ab  the  centrifugal  force  -^V^E,  and  by  ad  the 

weighty  of  any  molecule  situated  on  the  surface,  we  shall 
have  the  proportion 


whence 


ah  or^VxaE  :  be  orp  :  :  E  :  01, 


which  shows  that  the  distance  OI  is  constant  for  all  points 
of  the  surface  of  the  liquid,  and  that  consequently  this 


FiS.  86. 


202 


MOTION   OF   ROTATION. 


surface  is  that  of  a  cylinder,  with  a  circular  base  of  radius 
al,  whose  axis  is  parallel  to  that  of  the  wheel.  This  the- 
orem, for  which  we  are  indebted  to  M.  Poncelet,  serves 
as  the  basis  of  the  theory  which  this  illustrious  engineer 
has  given  upon  the  effects  of  water  in  bucket  wheels  with 
great  velocities.'* 

177.  Regulators  with  centrifugal  force. — The  action 
of  centrifugal  force  is  utilised  in  the  construction  of  an 
apparatus  called  a  Governor.  It  consists  principally  of  a 
vertical  spindle  AH,  (Fig.  87,)  which  receives  from  the 
machine  to  be  regulated  a  motion  of  rotation.  Upon  this 
spindle  are  suspended  two  rods  AP  and  AP',  jointed  at 
A  and  terminated  by  the  equal  weights  or  bobs  P  and  P'. 
At  the  two  joints  B  and  B'  of  the  rods  AP  and  AP7  are 
jointed  two  other  equal  rods  BC  and  B'O',  forming  with 


the  first  a  lozenge,  and  which  at  their  ends  C  and  C'  are 
also  jointed  with  a  collar  traversed  by  the  vertical  spin- 

*  See  Lemons  sur  1'hydraulique. 


MOTION   OF   KOTATION. 

die  with  which  it  turns,  having  at  the  same  time  a  motion 
of  translation,  in  the  direction  of  the  length  of  this  spindle. 
This  collar  has  a  yoke  in  which  is  fastened  the  fork  of  a 
lever  DE,  which  acts  upon  the  throttle  valves  for  steam,  or 
upon  any  other  piece. 

The  working  of  this  contrivance  is  readily  understood. 
By  the  effect  of  the  rotary  motion  of  the  vertical  spindle, 
the  balls  of  the  regulator  are  thrown  outwards  from  the 
axis,  and  so  raise  the  collar  a  certain  height.  If  the  ma- 
chine has  attained  and  preserves  its  normal  velocity,  the 
balls  and  the  collar  are  held  in  the  same  position,  because 
there  is  established  a  state  of  equilibrium  between  the 
centrifugal  force  and  the  weights  of  the  different  parts  of 
the  apparatus.  "When  the  velocity  increases,  the  centri- 
fugal force  increases,  tending  to  spread  outwards  the  balls 
and  to  raise  the  collar,  and  consequently  the  lever  DE. 
Inversely,  if  the  velocity  diminishes,  the  balls  approach 
the  spindle ;  the  collar  and  the  end  of  the  lever  DE  are 
lowered. 

Let  us  examine  the  mechanical  conditions  of  the  ac- 
tion of  this  apparatus,  and  first  suppose  the  collar  CC',  as 
well  as  the  rods  BC  and  B'C',  to  be  in  equilibrium  with 
the  lever  DE,  so  that,  neglecting  friction,  we  may  regard 
the  rods  AB  and  AB'  as  free  to  yield  to  the  centrifugal 
force  which  tends  to  separate  them,  and  to  the  weight  of 
the  balls  which  tends  to  bring  them  nearer  to  the  spindle. 

p 

The  centrifugal  force  of  each  ball  is  —V*  x  OP,  and  its 

y 

moment  in  relation  to  the  axis  of  joints  A  is 

?Y21xOPxAO. 
9 

The  moment  of  the  weight  P  of  each  ball  in  respect  to 
the  same  axis  is 

PxOP. 


204  MOTION  OF  EOTATION. 

Consequently,  the  condition  of  equilibrium  of  each  is 
-V*  x  AO=P,  whence  —  =^  5 

y  if  ^ 

which  shows  that  the  distance  of  the  balls'  separation  from 
the  spindle  depends  not  upon  their  weight,  but  solely 
upon  the  angular  velocity  of  rotation,  and  enables  us  to 
so  dispose  of  the  weight  of  the  balls  as  to  satisfy  other 
conditions. 

If  we  call  T  the  time  of  the  revolutions  of  the  balls 
around  the  vertical  spindle,  we  have  Y1T=27r=6.28, 
whence 


,=     >  and  consequently       ^ 


whence 


which  is  double  the  duration  of  oscillations  of  a  pendulum 
having  for  its  height  the  height  AO,  at  which  the  balls 
would  be  raised  to  the  normal  velocity. 

The  above  formula  enables  us  to  determine  approxi- 
mately the  height  AO  at  which  the  balls  are  raised  with  a 
given  velocity,  and  thus  to  establish  their  mean  position. 
It  gives,  in  fact, 


Thus  for  T=r 


AO=0.81517ft- 


AO=3.2606ft- 


In  this  calculation  we  have  neglected  the  weight  and 
the  centrifugal  force  of  the  rods  AB  and  AB'. 

The  preceding  remarks  are  not  sufficient  to  insure  the 
action  of  the  pendulum  as  a  regulating  apparatus,  since 
it  is  a  requisite  that  it  should  be  able  to  move  the  lever 
DE  and  the  parts  for  the  distribution  of  the  steam  or 


MOTION   OF  ROTATION.  205 

water,  upon  which  this  lever  operates,  or  in  other  terms, 
it  should  be  able  to  overcome  the  resistances  experienced 
in  the  motion  of  the  collar,  when  the  balls  are  separated 
or  brought  nearer  to  each  other.  These  resistances  can 
be  estimated  or  measured  when  the  apparatus  is  con- 
structed, and  if  we  call 

2Q  the  vertical  force  applied  to  the  collar  in  the  direc- 
tion of  the  vertical  spindle, 

Y/1=(l+7z-/)  Vt  another  angular  velocity,  greater,  for 
example,  than  the  mean  velocity  Yt  by  a  fraction  n'  of 
the  latter.  It  is  easily  seen  that  the  force  2Q  can  be  re- 
solved into  two  other  forces  parallel  and  equal  to  Q, 
applied  at  each  of  the  joints  B  and  B7,  and  that  then  we 
shall  have  for  the  equilibrium  corresponding  to  these  new 
conditions,  at  the  instant  of  its  being  broken,  the  relation 

?V'\  x  OP  x  AO=P  x  OP+Q  x  BO7. 

g 

Calling  a  the  distance  AB=AB7  and  5  the  length 
AP—  AP'  of  the  rods  to  the  centre  of  the  balls,  we  remark 
that 

I  :  a  :  :  OP  :  BO7,  whence  B07=|.  OP, 
and  consequently 

PV/2  n 

—  >.AO=P+Q.£ 
9  l 

We  have  previously  found  that  the  value  of  AO  cor- 
responding to  the  mean  position  of  the  balls  was 


the  above  relation  becomes,  then, 


206  MOTION   OF   ROTATION. 

whence  we  derive 


a 


so  long  as  n'*  is  very  small  compared  with  n'. 

We  also  see,  then,  from  these  considerations,  due  to  M. 
Poncelet,  that  there  exists  a  necessary  relation  between 
the  ratio  of  the  weight  of  the  balls  to  the  resistance  and 
the  degree  of  regularity  of  which  the  apparatus  is  suscep- 
tible. 

We  see,  also,  that  for  a  degree  of  regularity  desired  or 
considered  as  necessary  in  the  operation  of  the  machine, 
the  weight  of  the  balls  increases  proportionally  with  the 
resistance  which  the  collar  opposes  or  experiences.  Then 
for  example,  if  we  have  the  proportions  a=  0.666,  and  if 

we  have  n'=—  =0.02,  we  find 
oU 

P_    0.66 
Q~2^002~ 

so  that,  if  the  resistance  of  the  collar  was  only  22.05 
pounds,  the  weight  of  each  of  the  balls  should  be 


This  result  shows  that  this  apparatus  cannot  give  a 
great  degree  of  regularity  to  machines,  without  great 
dimensions  and  weights,  if  we  would  overcome,  directly 
by  the  collar,  considerable  resistances. 

It  is  from  a  disregard  of  these  circumstances,  that 
many  constructors  have  failed  in  the  establishment  of  this 
kind  of  regulators,  made  for  the  purpose  of  raising  sluice 
gates,  or  in  fixtures  for  the  distribution  of  steam.  This 
serious  inconvenience  may  be  avoided  ;  and,  with  this  sim- 
ple and  solid  apparatus,  we  may  obtain  a  proper  regulation 
by  arranging  it  in  the  following  manner,  which  I  will  de- 
scribe for  the  case  of  a  hydraulic  wheel. 


MOTION    OF   ROTATION. 


207 


178.  Distribution  of  a  Regulator  with  centrifugal 
force. — The  vertical  spindle  of  the  regulator  bears  a  coni- 
cal wheel  aa',  geared  with  a  wheel  W  of  the  same  diame- 
ter, fixed  upon  a  horizontal  axle  cc'^  which  consequently 
makes  the  same  number  of  turns  as  the  spindle  AH.  This 
axle  carries  two  wheels  with  gentle  friction,  toothed  with 


FIG.  88. 

conical  gearings  ee'  and  ff  provided  with  a  shield  with 
catches,  which  can  engage  with  the  collar  ddr  movable 
upon  the  axle  ccf  in  the  direction  of  its  length.  It  follows, 
from  this  disposition,  that  these  two  wheels  ee'  andj^'  do 
not  partake  of  the  motion  of  rotation  of  the  axle  <?</,  except 
they  are  connected  with  the  collar  ddf. 

These  two  wheels  are  both  geared  with  a  third  wheel 
gg'  of  the  same  diameter,  placed  at  the  extremity  of  the 
horizontal  axle  of  an  endless  screw  perpendicular  to  the 
axle  cc'. 

It  follows  from  this  arrangement  that,  when  the  con- 


208  MOTION   OF  ROTATION. 

necting  collar  drives  either  of  the  wheels  ee'  or  ff^  the 
wheel  gg'  and  the  endless  screw  turn  in  either  direction  ; 
the  office  of  the  conical  pendulum  is  confined  then  to 
sliding  the  connecting  collar  gg*,  from  right  to  left  or  left 
to  right,  according  as  the  motion  is  accelerated  or  re- 
tarded ;  and  the  only  resistance  it  has  to  overcome,  is  that 
of  the  friction  of  this  collar  upon  the  axle,  and  against  the 
catches,  which  is  quite  small,  and  differs  little  from  one 
wheel  or  motor  to  the  other.  The  principal  resistance, 
that  of  manoeuvring  the  gate,  is  surmounted  by  the  motor 
itself,  which  drives  the  endless  screw,  which  last,  by  prop- 
erly proportioned  gearings,  opens  or  closes  the  gates.* 

Another  advantage  of  this  disposition,  is  that  the  same 
model  of  the  regulator  may  answer  for  all  cases,  provided 
the  velocity  of  the  balls  of  the  vertical  spindle  is  also  the 
same. 

Many  regulators  of  this  kind,  established  for  the  man- 
ufacture of  ordnance,  have  given  satisfactory  results.  I 
give  the  principal  proportions  for  the  case  of  a  wheel  fitted 
with  plane  floats  in  a  circular  course  whose  wier  gate  is 
6.56ft-  in  width. 

The  horizontal  axle  CC  (Fig.  89)  receives  the  motion 
of  the  wheel  by  means  of  a  fixed  pulley  and  a  belt,  and 
at  the  normal  speed  of  the  machine  makes  48  turns  per 
minute.  It  is  consequently  the  same  with  the  vertical 
axle  of  the  regulator,  to  which  motion  is  transmitted  by 
the  conical  wheels  aa!  W  of  figure  88,  which  have  each 
24  teeth. 

Each  ball  weighs  46.75  pounds  =  P,  and  has  a  diame- 
ter of  .574ft- 

We  have  «=AB=AB/=:0.82ft-  (Fig.  87),  £=AP= 
AP'=1.59ft-  The  weight  of  the  rods  AP  and  AP'  is  6.5 
pounds ;  that  of  the  rods  BC^B'C'  is  2.47  pounds. 


*  Beautiful  specimens  of  governors  working  on  this  principle,  may  be  seen 
upon  the  admirable  Turbine  Wheels  constructed  by  U.  A.  Boyden,  at  Lowell, 
Massachusetts. 


MOTION   OF   ROTATION. 


209 


Experiment  has  given,  for  the  effort  to  be  exerted  hori- 
zontally upon  the  balls,  to  raise  the  collar  and  the  lever, 
an  intensity  of  79.39  pounds. 

The  three  bevelled  wheels,  ee'^ff^  ggr,  have  30  teeth 
each. 

The  endless  screw  has  a  single  thread. 

The  wheel  with  helicoidal  teeth  LL,  (Fig.  89,)  which 
it  drives,  has  90  teeth. 

The  pinion  MM  (Fig.  88)  borne  by  the  axle  of  this 
wheel  has  16  teeth. 

The  wheel  NN  which  this  pinion  drives  has  80 
teeth. 

The  pinion  of  the  rack  mounted  upon  the  same  axle 
has  24  teeth. 

The  pitch  of  the  cogs  of  the  rack  is  .075" 


FIG.  89. 


It  follows,  from  these  proportions,  that  calling  K  the 
number  of  turns  of  the  vertical  axle  and  that  of  the  end- 
14 


210  MOTION   OF   ROTATION. 

less  screw,  which  -are  the  same,  that  for  the  axle  of  the 
pinion  will  be 

1ST     16=__N_  _JN_ 
90X80~90x5~450' 

If  we  suppose  IS"—  10,  the  pinion  which  has  24:  teeth  will 
raise  the  rack,  whose  cogs  are  .075ft-  apart. 


450 

Thus,  for  the  turns  of  the  vertical  spindle  of  the  regu- 
lator, the  gate  is  raised  or  lowered  0.04ft- 

This  ratio  between  the  lifts  of  the  gate  and  the  number 
of  turns  of  the  spindle  of  the  regulator  is  important  to 
note,  and  in  any  applications  of  this  apparatus  which  it  is 
desired  to  make,  it  will  be  well  to  follow  with  similar 
wheels,  for  reasons  which  we  will  point  out  in  E"o.  181. 

179.  Results  of  observations  made  upon  the  effect  of  this 
regulator.  —  The  length  of  the  movable  collar  ddf  being 
less  than  the  space  between  the  wheels  ee'  andjg^,  it  fol- 
lows, that  so  long  as  the  collar  does  not  engage  with 
either  of  the  wheels,  the  apparatus  is  indifferent  to  the 
variations  of  the  velocity.  In  fact,  we  see  that  it  is  neces- 
sary to  maintain  a  certain  latitude  in  this  respect.  The 
mean  velocity  between  these  limits  is  the  normal  velocity 
of  the  wheel,  which  is  nearly  9.8  turns  in  1  minute. 

"When  the  velocity  is  reduced  to  9.5  turns  a  minute, 
the  grooves  of  the  collar  are  on  the  point  of  engaging  with 
those  of  the  wheel  ee'  ;  when  the  velocity  is  raised  to  10.1 
turns  in  1  minute,  the  grooves  of  the  collar  are  on  the 
point  of  engaging  with  those  of  the  wheel  ff.  Between 
the  limits  of  9.5  and  10.1  turns  in  1  minute,  the  velocity 
may  change  without  any  action  of  the  regulator.  The 
mean  of  these  velocities  is  9.8  turns  per  minute.  The 
deviation  above  or  below  the  mean  velocity  is  then  0.3 


MOTION   OF  ROTATION.  211 

turns  in  1  minute,  or  --— -  of  the  mean  velocity  before  the 
o2.o 

commencement  of  the  action  of  the  regulator. 

When,  by  an  elevation  of  the  level,  or  by  a  diminu- 
tion of  the  resistance,  the  velocity  of  the  wheel  exceeds 
10.1  turns  per  minute,  the  collar  engages  with  the  wheel 
ffi  which  then  drives  the  wheel  gg'  and  the  endless  screw 
so  as  to  raise  the  gate  and  diminish  the  velocity.  For 
restoring  this  velocity  to  its  normal  value>  the  action  of 
the  regulator  must  have  a  certain  duration.  In  closing 
suddenly  all  the  gates  of  the  mills  upon  the  same  canals, 
for  obtaining  a  rapid  rise  of  the  levels,  the  velocity  may 
be  increased  to  11.1  turns  in  1  minute.  The  maximum 
departure  above  the  mean  velocity  thus  produced  has 

been  as  high  as  1.3  turns  in  1  minute,  or  — -  of  the  mean 

T  .5 

velocity,  and  it  requires  from  50  to  60  seconds'  action  of 
the  collar  to  regain  the  velocity  of  10.1  turns  in  1  minute, 
at  which  it  is  disengaged. 

When,  by  reason  of  an  increased  resistance,  or  a  low- 
ering of  the  level,  the  velocity  of  the  wheel  has  reached 
9.5  turns  in  1  minute,  the  collar  engages  with  the  wheel 
eef,  which  then  drives  the  wheel  gg'  and  the  screw  in  an 
opposite  direction,  and  lowers  the  gate,  and  so  restores  the 
velocity  to  its  mean  value. 

On  lowering  suddenly  the  level  in  the  upper  reach, 
the  velocity  of  the  wheel  may  be  as  low  as  8  turns  per 
minute,  which  corresponds  to  a  departure  of  1.8  turns  per 

minute  of  — -  of  the  mean  value.     But  by  the  action  of 
5.5 

the  regulator,  the  wheel  is  restored  to  the  velocity  of  9.5 
turns  per  minute  at  the  end  of  30  seconds. 

We  see  by  these  details  that  this  regulator,  by  the 
delicacy  of  its  operations,  may  be  employed  very  usefully 
in  many  cases. 


212  MOTION   OF  ROTATION. 

180.  Comparison  of  the  data  of  experiment  with  the 
formula. — The  results  of  direct  observations  made  upon 
the  governors  at  Bouchet's  powder-mills,  give  us  the  means 
of  verifying  the  exactness  of  the  formula  of  No.  1YT. 

We  have  seen,  beyond  a  velocity  of  10.1  turns  in  1 

minute,  exceeding  the  mean  velocity  of  — ,  that  the  collar 

32 

engages  with  the  wheel  ff,  to  check  the  speed,  and  is 
afterwards  disengaged,  when  from  a  gain  of  speed  it  tends 
to  return  to  its  normal  value. 

Now,  the  normal  velocity  of  the  regulator  is  48  turns 

of  its  vertical  spindle,  and  for  an  increase  of  — =n',  itbe- 

32 

comes  49  turns  per  minute,  which  corresponds  to  an  angu- 
lar velocity  of 

fi  28 

'-  x49.5=5.18ft-  per  second. 


We  have  then  OP=1.062ft-  nearly. 

Consequently,  the  centrifugal  force  of  each  ball  be- 
comes 

p  4.fi  n** 

-  Y:  x  OP=-^?-  x  (5.18)8  x  1.062ft-=41.49lb% 


which,  for  the  balls,  gives  us  an  effort  of  82.98lb%  very 
nearly  equal  to  that  indicated  by  direct  observations  for 
raising  the  collar  and  the  gearing  lever. 

181.  Modification  of  the  balls  for  obtaining  a  greater 
regularity.  —  If  we  wish  to  impart  a  greater  sensibility  to 
the  machinery,  it  will  suffice  to  increase  the  balls,  know- 
ing that  for  the  regulator  in  consideration,  the  effort  to  be 
exerted  by  the  balls  is  82.98lbs-  or  41.49lbs-  for  each.  If 

the  degree  of  regularity  n'  is  to  be  —  -  of  the  mean  veloci- 

50 


MOTION   OF   ROTATION.  213 

ty,  the  number  of  turns  of  the  axle,  at  the  commencement 
of  the  action  of  the  balls,  should  be 

48+g=48.96; 
consequently, 

V^—  x  48.96 =5.1244ft- 

Admitting  that  OP=1.062f%  we  shall  have  for  the  de- 
termination of  the  new  weight  P  of  the  balls 


whence 


x  (5.124)2  x  1.062=41.49% 


..=  g> 

(5.124)2x  1.062 


182.  Observations  upon  the  transmission  of  motion  ty 
the  endless  screw  to  the  gate. — In  the  first  applications  of  this 
kind  of  regulators,  it  was  observed  that  their  play  was  in- 
cessant, and  that  the  gate  was  raised  or  lowered  continu- 
ally, so  that  the  action  of  the  wheel  was  quite  irregular, 
and  was  far  beyond  or  short  of  its  mean  velocity.  This 
inconvenience,  experienced  also  in  many  other  applica- 
tions, has  caused  constructors  to  regard  the  governor  as 
defective,  and  more  injurious  than  useful  for  regularity 
of  motion.  But  on  examining  more  carefully  its  action, 
and  that  of  the  gates,  I  discovered  that  the  ceaseless  vari- 
ations arose  from  imparting  too  great  motion  to  the  gate, 
so  that  it  rose  and  fell  too  suddenly,  and  always  beyond 
the  adopted  limit  of  mean  action  ;  and  thus  irregularities, 
inherent  upon  the  mode  of  action  of  the  machine  upon 
variations  in  the  level  of  the  water,  &c.,  were  constantly 
added  to  the  already  exaggerated  motion  of  the  gates.  I 
changed  the  motion  of  transmission  to  the  gates  so  as  to 
reduce  considerably  the  motion  of  the  latter,  and  thus 


214  MOTION   OF   ROTATION. 

confined  these  variations  of  velocity  within  convenient 
limits. 

Similar  observations  were  made  at  one  of  the  great 
sharpening  establishments  at  Chatellerault,  upon  the  effect 
of  a  regulator  applied  to  a  turbine ;  and  the  working  of 
this  motor  has  been  very  well  regulated  by  establishing 
similar  proportions  for  the  transmission  of  motion  of  the 
screw  to  the  gate. 

According  to  these  observations,  it  was  admitted,  that 
when  the  vertical  spindle  of  the  regulator,  and  that  of  the 
endless  screw,  were  regulated  to  a  mean  velocity  of  48 
turns  in  1  minute,  the  gates  of  the  weirs  for  the  side 
wheels  should  not  be  raised  or  lowered  more  than  from 
.033  to  .049  ft-  for  10  turns  of  the  endless  screw,  and  that 
for  turbines,  Fontaine's,  for  example,  the  run  of  the  gates 
should  not  exceed  from  .005  to.006ft-  for  10  turns  of  the 
screw. 

In  satisfying  these  conditions,  by  suitable  proportions 
in  the  transmission  of  motion  from  the  screw  to  the  gate, 
the  deviations  of  velocity  were  confined  within  sufficient 
limits. 

183.  Indispensable  disposition  in  the  use  of  these  regu- 
lators.— The  gate  of  the  hydraulic  motor  to  which  is  ap- 
plied this  kind  of  regulator  being  driven  by  the  motor 
itself,  we  see  that  for  gates  in  weirs,  which  are  to  be  low- 
ered when  the  motion  slackens,  and  for  gates  with  water 
upon  their  summits,  which  are  to  be  raised  when  the 
resistance  increases,  the  run  of  these  organs  being  neces- 
sarily limited,  it  is  important  to  stop  the  action  of  the 
regulator  before  these  limits  are  attained,  else  some  rup- 
ture might  occur. 

It  is  necessary  then  to  arrange  some  disengaging  con- 
trivance, to  interrupt  the  action  of  the  endless  screw  upon 
the  gate,  as  soon  as  it  is  upon  the  point  of  attaining  the 
limit  of  its  course. 


MOTION   OF  ROTATION.  215 

184.  Modification  of  the  apparatus  just  described. — To 
render  this  apparatus  more  sensible  to  the  small  variations 
in  velocity,  M.  Delongchamp,  Civil  Engineer,  proposed  to 
transmit  the  motion  to  conical  wheels,  by  means  of  a  belt, 
passing  from  one  pulley,  always  loose,  over  two  others 
placed  on  the  right  and  left  of  it,  which  were  fastened 
upon  the  axle  of  bevelled  wheels.     The  action  of  the  pen- 
dulum was  then  reduced  to  passing  the  belt  from  one 
pulley  to  the  other,  which  only  required  a  small  effort, 
and  avoided  the  shock  of  the, clutches  of  the  collar,  in  the 
other  disposition.     Thus  modified,  one  regulator  might 
answer  for  a  great  number  of  different  cases. 

185.  Other  regulators. — There  are  other  contrivances 
constructed  for  the  same  purpose  as  the  preceding,  which 
we  have  mentioned  only  to  show  an  example  of  the  cen- 
trifugal force.     In  passing,  we  will   only  allude   to   the 
Molinie  regulator,  that  of   M.   Lariviere,  and  that  of 
Siemens,  based  upon  the  use  of  the  conical  pendulum  of 
Huyghens,  &c.     Good  results  may  be  obtained  by  them, 
but  this  is  not  the  place  to  discuss  them. 

186.  Variable  motion  around  an  axis. — We  have  seen 
from  what  precedes,  that  in  the  motion  of  rotation  about 
an  axis,  the  work  of  all  the  external  forces  soliciting  the 
body  is  equal  to  the  work  of  their  resultant.     We  may 
then  consider  all  these  forces  as  replaced  by  this  resultant. 

If  the  motion  is  uniform,  it  is  evident  that  the  work 
of  all  the  forces  tending  to  accelerate  the  motion  will  be 
equal  to  that  of  the  forces  tending  to  retard  it,  or  that 
the  work  of  the  resultant  will  be  zero.  But  if  the  work 
of  this  resultant  is  not  zero,  there  must  necessarily  be  pro- 
duced a  certain  variation  in  the  velocity  of  the  body,  and 
then  the  inertia  of  each  of  the  elementary  masses  com- 
posing it  develops  in  an  opposite  direction,  efforts  propor- 
tioned to  the  degrees  of  velocity  either  imparted  to  or 
taken  from  it. 


216  MOTION   OF   ROTATION. 

Let  us  call  V,  the  angular  velocity,  or  the  circular 
space  described  by  a  point  at  a  unit  of  distance  from  the 
axes,  during  a  unit  of  time,  so  that  at  the  instant  consid- 
ered the  motion  may  be  uniform,  and  vr  the  elementary 
variation  which  this  velocity  expe- 
riences in  the  element  of  time  t,  any 
elementary  mass  m  situated  at  a  dis- 
tance  r  will  be  impressed  with  the 
velocity  Y^,  and  the  elementary  va- 
riation of  this  velocity  will  be  v^r. 
Consequently,  inertia,  by  its  reaction, 
will  develop  in  a  direction  opposite 
to  the  work  of  the  resultant  of  the 

exterior  forces,  an  effort  m—  directed  tangential  to   the 

t 

circumference  described  by  the  mass  m.  Now,  if  to  each 
of  these  elementary  masses  m  is  applied,  in  a  direction 
opposite  to  the  variation  of  motion,  a  force  equal  to 

0)  />* 

m— ,  and  in  the  direction  of  the  reaction  of  inertia,  this 
t 

force  will  be  able  to  destroy  the  variation  of  velocity  sy, 
and  consequently  the  effect  of  the  general  resultant  of  the 
external  forces  upon  the  mass  m  ;  then  these  forces  combined 
will  destroy  the  effects  of  the  general  resultant  of  the  ex- 
terior forces,  and  consequently  they  will  be  in  equilibrium. 
Now,  these  forces  which  we  have  supposed  applied  to 
each  of  the  molecules  of  the  body,  are  precisely  equal  to 
the  reactions  developed  by  inertia,  and  have  the  same 
direction.  There  is  then  also  at  each  instant  of  the  varia- 
tion of  motion,  an  equilibrium  between  the  reactions  and 
the  resultants  of  the  exterior  forces,  or,  what  amounts  to 
the  same,  the  work  developed  by  all  the  reactions  must 
be  equal  to  the  work  developed  by  the  resultant. 

The  elementary  arc  described  by  the  mass  m  being 
a^r,  calling  av  the  elementary  arc  at  the  unit  of  distance, 


MOTION   OF  KOTATION. 


217 


the  work  developed  by  the  force  of  inertia  during  the 
variation  of  the  velocity  will  be  for  the  mass  m 

v.r 

m.-L.a?. 

Now,  —  ^Yj?1,  or  the  velocity  possessed  by  the  mass 
t 

m  at  the  instant  considered ;  then  the  elementary  work 
of  the  force  of  inertia  of  the  mass  m  has  for  its  expression 

mv.r  x  V1r=mriVlvl. 

The  product  of  the  mass,  by  the  square  of  its  distance  r 
from  the  axis  of  rotation,  entering  in  this  expression,  is 
called  the  moment  of  inertia  of  this  mass. 

For  another  elementary  mass  w',  situated  at  a  distance 
/,  we  shall  have  for  the  work  of  inertia  mV^V^,  and  for 
any  similar  number  of  masses  of  elementary  work,  the 
sum  of  the  quantities  developed  by  their  inertia  will  be 


V  "  +  &c. 


187.  Important  observations  upon  the  moments  of  iner- 
tia. —  Geometry  teaches  us  how  to  calculate  the  sums  of 
the  moments  of  inertia  of  the  elements  of  different  formed 
bodies,  as  we  shall  show  further  on  ;  but,  for  the  present, 
it  is  well  to  explain  an  im- 
portant theorem  as  to  the 
moment  of  inertia  of  a  body 
in  its  relation  to  any  axis 
when  the  moment  of  its  in- 
ertia with  respect  to  a  paral- 
lel axis  passing  through  the 
centre  of  gravity  of  the  body 
is  known. 

Let  us  consider  an  ele- 
mentary mass  m  of  a  body 
whose  centre  of  gravity  is  G,  FIG.  91. 


218  MOTION   OF   ROTATION. 

and  which  turns  around  an  axis  A.     The  moment  of  iner- 
tia of  this  element,  in  respect  to  the  axis  A,  will  be 


But  if  we  call  AG^^Z  the  distance  of  the  centre  of  grav- 
ity from  the  centre  of  rotation,  and  Gm=r1  the  distance 
of  the  molecule  from  the  centre  of  gravity,  we  have  by 
the  triangle  Aaw,  found  by  letting  fall  ma  upon  AG  pro- 
duced, 

Am8  =  K.c?-\-fmc£<) 
or  since 

Aaa=AG2+2AG  x 


or 

r* 

The  moments  of  the  mass  m  is  then 

%  .md. 


a  formula  in  which  we  remark  that  m  .  aG  is  the  moment 
of  the  mass  m,  in  respect  to  a  plane  perpendicular  to  the 
line  AG,  and  passing  through  the  centre  of  gravity.  So 
for  the  other  masses  m',  m",  &c.,  situated  at  the  distances 
r',  r",  r"f  from  the  axis  A,  and  at  distances  r/,  71/',  /•/", 
from  the  centre  of  gravity,  we  shall  have 

'd.  a'G, 


And  consequently,  calling  I  the  total  amount  of  inertia  in 
respect  to  the  axis  A,  and  I1=m/*18+mV/24-m/V//2+&c., 
the  moment  of  inertia  in  relation  to  the  parallel  axis  pass- 
ing through  the  centre  of  gravity  of  G  and  M,  the  total 
mass  of  the  body  =  m-\-mf+m"+&G.)  we  have 


MOTION   OF   EOTATION.  219 

Now,  the  term  in  the  parenthesis  is  the  sum  of  moments 
of  the  elementary  masses  composing  the  body,  in  respect 
to  a  plane  passing  through  the  centre  of  gravity  ;  it  is 
then  zero,  and  the  above  relation  is  reduced  to 


which  expresses  that  the  moment  of  inertia  of  a  body,  in 
respect  to  any  axis,  is  equal  to  the  moment  of  inertia  of 
the  same  ~body  in  relation  to  an  axis  parallel  to  the  first, 
and  passing  through  the  centre  of  gravity  of  the  body  ',  plus 
the  product  of  the  mass  of  the  body,  by  the  square  of  the 
distance  of  the  two  axes. 

188.  Principle  of  vis  viva  in  the  mot^on  of  rotation 
about  an  axis.  —  It  follows,  from  what  has  been  said  in  'No. 
186,  that  in  calling  I  the  moment  of  the  total  inertia  of 
the  body  in  consideration,  the  work  developed  by  inertia 
during  the  elementary  valuation  vl  of  the  angular  velocity 
will  be  1^^  and  we  have  seen  this  quantity  should  be 
equal  to  the  work  developed  in  the  same  time,  by  the  re- 
sultant of  the  exterior  forces  ;  which  otherwise  may  be 
apparent,  in  observing  that,  if  the  work  of  forces  of  inertia 
was  inferior  to  that  of  the  resultant,  in  taking  the  first 
from  the  second,  the  excess  would  produce  an  acceleration 
or  reduction  of  the  velocity  other  than  that  which  really 
takes  place. 

We  have,  then,  at  any  instant, 

I.V^^K.a^, 

in  calling  E  the  resultant  of  all  the  external  forces,  and  r, 
its  arm  of  lever. 

At  the  end  of  a  certain  time,  the  work  of  this  variable 
K,  variable  or  constant,  may  be  obtained,  either  directly 
or  by  Simpson's  method,  and  may  be  represented  by  W. 

As  for  the  total  work  of  the  forces  of  inertia,  the  factor 
I  depending  solely  upon  geometrical  dimensions,  and  the 
material  of  which  the  body  is  composed,  the  sum  of  all 
the  similar  quantities  of  work,  from  the  instant  or  the  po- 


220  MOTION   OF  ROTATION. 

sition  where  the  angular  velocity  is  Y1?  to  that  where  it 
has  reached  Y/,  will  be,  from  what  we  have  previously 
seen,  represented  by 


if  the  motion  is  accelerated,  R  being  a  motive  force,  or  by 


if  the  motion  is  retarded,  R  being  a  resistant. 

Consequently,  at  the  end  of  any  time,  when  the  angu- 
lar velocity  shall  have  passed  from  the  velocity  Yj  to  the 
value  Y/,  we  shall  have  between  the  quantities  of  work 
developed  by  the  external  forces  or  their  resultant,  and 
by  the  forces  of  inertia,  the  relation 


We  would  remark  that,  I  being  the  sum  of  the  elementary 
products  rar2,  mVa,  &c.,  we  have 


an  expression  in  which  mr*V*,  mVYj2,  are  evidently 
what  we  have  hitherto  called  the  vis  viva  of  the  masses 
m,  mf,  &c.  ;  then  lY^,  IY/2,  are  the  sums  of  the  vis  viva 
of  the  body,  and  the  above  relation  shows  us  that,  in  the 
motion  of  rotation,  as  well  as  in  that  of  translation^  the 
work  developed  by  the  exterior  forces  at  the  end  of  a  cer- 
tain time,  is  equal  to  the  half  of  the  vis  viva  acquired  or 
lost  ~by  the  body  during  the  same  time. 

We  see,  then,  that  the  principle  of  vis  viva,  previously 
demonstrated  for  the  parallel  motions  of  translation,  is  also 
true  for  the  motions  of  rotation  about  an  axis. 

Now,  as  any  elementary  motion,  velocity  or  work  can 
always  be  decomposed  into  two  elementary  motions, 
velocities  or  works,  the  one  of  translation  in  the  direction 


MOTION   OF  ROTATION. 


221 


of  a  certain  axis,  the  other  of  rotation,  perpendicular  to 
this  same  axis,  and  as  in  this  decomposition  the  square  of 
the  resultant  velocity  is  equal  to  the  sum  of  the  squares 
of  the  component  velocities,  and  as  the  sum  of  the  com- 
ponent living  forces  is  equal  to  the  resultant  vis  viva,  and 
as  the  resultant  work  is  equal  to  the  sum  of  the  compo- 
nent works,  it  follows,  evidently,  that  in  any  motion  the 
work  developed  at  the  end  of  a  certain  time  by  exterior 
forces  is  equal  to  half  of  the  variation  of  the  correspond- 
ing vis  viva  during  the  same  interval. 

Such  is  the  enun- 
ciation of  the  princi- 
ple of  the  vis  viva  in 
its  most  general  form, 
and  it  serves  as  a 
base  for  the  general 
theory  of  machines 
and  of  the  motions  of 
bodies. 

Before  applying 
this  principle  to  the 
motion  of  machines, 
we  will  make  use  of 
it  in  the  study  of  the 
motion  of  pendulums 
and  of  ballistic  pen- 
dulums in  particular. 

189.  Theory  of  the 
Pendulum. — For  a 
first  application  of  the 
preceding  principles, 
we  will  attend  to  the 
theory  of  the  pendu- 
lum; and  first  suppose 
that  we  consider  the  FIG.  92. 

motion  of  an  elementary  mass  suspended  to  an  infinitely 


222  MOTION   OF  ROTATION. 

thin  wire,  and  search  out  the  various  conditions  of  the 
motion  of  this  contrivance,  which  we  term  a  simple  pen- 
dulum, and  which  would  also  be  found  very  nearly  in  the 
same  circumstances  with  a  lead  ball  suspended  upon  a 
very  fine  silk  thread  supposed  to  be  rigid. 

Suppose  the  pendulum,  starting  from  the  point  B,  has 
arrived  to  M,  and  has  consequently  fallen  a  height 
MP=H,  the  work  developed  by  gravity  will  be  mgH9 
and  if  we  call  Y  the  velocity  of  the  mass  m,  in  the  direc- 
tion of  the  tangent  to  the  circle  described,  its  vis  viva  will 
be  mY3,  and,  according  to  the  principle  of  the  vis  viva, 
we  shall  have 


or  v  =: 
The  velocity  Y  of  this  variable  motion  has  also  for  its 

o 

expression  the  ratio  -  of  the  elementary  arc  described  in 

the  element  of  time  i;  the  above  relation  is  then  re- 
duced to 

sa  °a 

— — 2^H;  whence  f= 

or 

't= 


If  we  compare  this  pendulum,  whose  length  is  AB=/*, 
with  another  whose  length  is  AB'=/,  which  describes  an 
equal  angle,  and  is  placed,  when  the  velocity  imparted  to 
heavy  bodies  in  the  first  second  of  their  fall  is  g ',  we  shall 
also  have 


We  shall  have,  then,  for  these  two  pendulums  the  pro- 
portion 

S*  «'* 

fit"::  ^ 


MOTION   OF  ROTATION.  223 

But  the  condition  that  the  angle  described  by  two 
pendulums  may  be  equal  give  us  for  the  same  elementary 
angular  displacement, 

s  i  s'  :  :r:r',  or  8*  :  s'*  :  :  r*  :  /2, 
and,  moreover,  we  have 

H  :  H'  :  :  r  :  /, 


whence 
consequently, 


*   .  JL  •  •  r  •  rf 
H'H"  ' 


. 

-       ..-.-„ 

whence 


ct  w* 

We  would  observe  that  the  ratios  y-  and  -,  being  giv- 

t/ 

en  and  independent  of  the  described  angles,  it  follows 
that  the  elementary  times,  employed  to  describe  the  ele- 
mentary arcs  s  and  s',  are  in  constant  ratio,  and  that  con- 
sequently it  is  the  same  for  the  sum  of  the  elementary 
times  of  the  total  times  T  and  T'  employed.  in  describing 
an  entire  oscillation.  We  have  then,  also, 


g 

Such  is  the  relation  between  the  times  of  the  oscilla- 
tions of  simple  pendulums  in  different  places,  and  for 
different  lengths. 

If  we  compare  pendulums  of  the  same  length,  we  have 
r=rf,  and  then 


224  MOTION   OF  ROTATION. 

which  shows  that  the  duration  of  oscillations  of  pendu- 
lums of  the  same  length,  at  different  places  of  the  earth, 
are  to  each  other  in  the  inverse  ratio  of  the  square  roots 
of  the  values  of  g,  and  may  serve  to  determine  the  latter. 
At  the  same  place  we  have  g=g',  and  then 


whence  it  follows  that  then  the  times  of  oscillations  are 
to  each  other  as  the  square  roots  of  the  lengths  of  pendu- 
lums, as  Galileo  had  discovered,  by  direct  observation, 
before  the  making  of  the  theory. 

190.  Time  of  oscillations  of  a  pendulum  with  small 
vibrations.  —  If,  in  the  relation 

t_  _  «_ 


we  seek  to  introduce  the  value  of  the  elementary  arc  $, 
described  in  the  instant  t,  as  a  function  of  the  data  of  the 
figure,  we  have  by  the  similar  triangles  MQ1ST  and  MAF, 
(Fig.  92,) 

MN  :  QN  :  :  AM  :  ME,  or  *  :  QN  :  :  r  :  ME; 
whence 


'ME" 

Now,  ME  is  a  mean  proportional  between  CE  and  2r— CE, 
2r  being  the  diameter  of  the  circle  described  by  the  pen- 
dulum ;  we  have,  then, 


E=  V2r  x  CE-CE2. 

But  when  the  amplitude  of  oscillation  is  very  small,  we 
may  neglect  the  square  of  CE,  or  the  sagitta  of  the  de- 
scribed arc,  in  its  relation  to  the  product  2r  x  CE,  which 
reduces  the  above  value  to 


MOTION  OF  ROTATION.  225 

We  have  then 


t_     s  QN.r 


whence 


'  CE 


But  if  we  describe  upon  CD  as  a  diameter  a  circle, 
and  if  we  draw  the  parallels  Mm  and  !NV&  to  the  chord  BD, 
we  shall  have 

^E2=CE  xDE=CE  xH, 
which  gives 


E 
Now  the  similar  triangles  mOE  and  mqn  give 

£7i  or  QN  :  mE  :  :  mn  :  mO  ; 
whence 

QN_  mn 

and  consequently 
whence 


We  see,  then,  that  the  infinitely  small  time  employed  by 
the  pendulum  in  describing  the  elementary  arc  MN  is 
equal  to  the  product  of  the  constant  factor 


_    __ 
g'mtf 

by  the  element  mn  of  the  circumference  of  the  circle  de- 
scribed upon  CD  as  a  diameter.  Then  the  sum  of  all  the 
elements  of  time  successively  employed  in  describing  the 
arc  BC  will  be  equal  to  the  same  factor  multiplied  by  the 
15 


226  MOTION   OF   ROTATION. 

semicircle  of  which  DC  is  the  diameter  or  mO  the  radius, 
which  is  equal  to  7r7^O=3.14.  mO  ;  we  shall  have  then, 
for  the  total  duration  of  the  semi-  oscillation, 


g'     mO 
and  for  the  entire  oscillation 


Such  is  the  formula  which  gives  the  time  of  oscillation 
of  the  simple  pendulum.  From  this  we  deduce  the  ve- 
locity imparted  to  heavy  bodies  by  gravity  in  the  first 
second  of  their  fall, 


which  shows  how  the  knowledge  of  the  duration  of  small 
oscillations  of  a  simple  pendulum  of  known  length  may 
serve  to  determine  the  value  of  the  number  g. 

But  the  simple  pendulum  is  only  an  abstraction,  and 
is  only  used  as  an  approximate  method  for  measuring 
time  by  the  duration  of  the  oscillations  of  lead  balls  or 
other  heavy  bodies  suspended  upon  a  thread,  which  appa- 
ratus is  considered  as  a  simple  pendulum,  the  mass  of 
which  is  collected  at  the  centre  of  its  figure. 

In  ordinary  cases,  for  the  pendulums  of  clocks,  and 
with  still  better  reasons  for  those  employed  in  the  determi- 
nation of  the  velocities  impressed  by  powder  upon  pro- 
jectiles, which  are  therefore  called  ballistic  pendulums, 
we  must  take  into  account  the  distribution  of  the  mass. 

191.  The  compound  pendulum.  —  Let  us  consider,  then, 
a  solid  body  turning  or  oscillating  about  a  fixed  axis,  and 
regard  the  various  conditions  of  its  motion. 


MOTION   OF   ROTATION.  227 

Calling,  as  we  have  already  done,  I  the  moment  of 
inertia  of  the  body  in  respect  to  its  axis  of  rotation,  and 
V1  the  angular  velocity  at  an  instant  when  its  centre  of 
gravity  has  fallen  the  height  H,  we  shall  then  have,  by 
the  principle  of  vis  viva, 


and  if  we  call  d  the  distance  of  the  centre  of  gravity  of 
the  pendulum  from  the  axis,  and  Hj  the  height  which  a 
point  at  the  unit  of  distance  has  fallen,  the  proportion 

H,  i  lft-  :  :  H  :  d, 
whence  11=11^,  and  the  above  relation  becomes 


This  relation  is  of  the  same  form  as  that  presented  by 
the  simple  pendulum,  and  only  differs  in  the  factors  -y- 

which  depends  solely  upon  the  dimensions  and  the  nature 
of  the  body. 

We  have  also  for  the  angular  velocity  Ya=  -,  which 

t 

leads  to  the  relation 


Keasoning  here  precisely  as  we  have  done  for  the  sim- 
ple pendulum,  and  supposing  the  amplitude  of  oscillation 
very  small,  which  admits  of  our  neglecting  the  influence 
of  the  resistance  of 'the  air,  we  shall  see  that  the  fraction 


mo  mo 


228  MOTION   OF  ROTATION. 

by  reason  of  ^=1"-  ;  whence  it  follows  that  the  duration 
of  an  elementary  fraction  of  an  oscillation  has  for  ex- 
pression 


2     ~M.dg     mo  ' 
and  that  the  total  duration  of  the  oscillation  is 


T='  . 

.dg 

192.  Length  of 'the  simple  pendulum  which  makes  its 
oscillations  in  the  same  time  as  the  compound  pendulum. — 
If  we  compare  the  formula  of  the  simple  pendulum  with 
that  of  the  compound,  we  see  that,  in  order  that  the  times 
of  oscillations  may  be  equal,  we  must  have 


p 

which  gives  for  the  length  sought  of  the  simple  pendulum 

I 


193.  Determination  of  the  moment  of  inertia  of  a 
compound  pendulum. — When,  in  the  formula 


we  know  the  total  mass  of  the  pendulum,  and  the 
distance  d  of  its  centre  of  gravity  from  the  axis  of  knife 
blades,  or  the  suspension,  observation  of  the  duration  T 
of  the  oscillations  will  give  for  the  moment  of  inertia  in 
respect  to  the  axis 


MOTION  OF  ROTATION. 


229 


in  which  we  may  dispense  with  the  calculation  (quite 
laborious  in  many  cases)  of  the  moment  of  inertia. 

This  formula  will  be  found  peculiarly  applicable  in 
the  determination  of  the  moments  of  inertia  of  fly-wheels, 
ballistic  pendulums,  &c.  It  will  suffice  to  make  them 
oscillate  around  any  axis,  placed  at  a  known  distance 
from  their  centre  of  gravity,  in  drawing  them  slightly 
from  the  vertical,  and  observing  the  duration  of  their 
oscillations  by  counting  their  number. 

Let  us  farther  bear  in  mind  (No.  187)  that  in  calling 
I,  the  moment  of  inertia  in  respect  to  an  axis  passing 
through  the  centre  of  gravity,  and  parallel  to  the  axis  of 
suspension,  we  have  the  relation, 


which  gives,  at  our  need,  the  moment  of  inertia  in  respect 
to  an  axis  passing  through  the  centre  of  gravity. 

It  follows,  also,  that  the  length  of 
the  simple  pendulum  which  makes 
its  oscillations  in  the  same  time  as 
the  compound  pendulum,  and  which 
we  will  designate  by  &,  has  for  ex- 
pression 

,_  I       , 
~~   + 


-Hi 

and  that  it  is  always  greater  than 

that  of  the  centre  of  gravity  from 
the  axis. 

In  placing  upon  the  line  AG, 
which  joins  the  axis  to  the  centre    n 
of  gravity,  a  length 

-^i. 


FIG.  93. 


all  the  points  which  can  be  found  upon  the  line  parallel 


230  MOTION   OF   KOTATION. 

to  the  axis,  drawn  through  the  point  O,  may  be  regarded 
as  the  centres  of  so  many  simple  pendulums,  whose  oscil- 
lations are  made  in  the  same  time  with  those  of  the  com- 
pound pendulums.  This  point  O,  thus  determined,  is 
called  the  centre  of  oscillation  of  the  pendulum. 

It  is  well  to  remark  that  the  point  A  will  be  recipro- 
cally the  centre  of  oscillation  of  the  same  pendulum  if  the 
centre  O  becomes  the  point  of  suspension.  In  fact,  if  we 
call  df  the  distance  OGr  of  the  centre  of  gravity  from  the 
point  O,  we  shall  have  for  the  distance  W  of  the  new  cen- 
tre of  oscillation  from  the  axis  O, 


But 

whence  we  derive 


and  consequently 

Jc'=7c— d+d=L 

194.  Determination  of  the  centre  of  gravity  of  corn- 
pound  pendulums. — This  operation  is  done  by  calculation, 
or  by  the  means  pointed  out  in  No.  14:2,  or  by  a  combi- 
nation of  the  two  methods.  Sometimes,  for  ballistic  pen- 
dulums whose  weight  may  reach  several  thousands  of 
pounds,  we  adopt  the  following  method : 

We  fasten  to  the  cannon  or  receiver,  at  any  point  of 
their  suspension,  a  cord  passing  over  a  pulley,  on  which 
we  hang  a  weight,  which  holds  the  pendulum  at  a  de- 
terminate inclination.  The  pulley  should  be  large,  its 
axle  small  and  well  oiled,  so  that  friction  may  be  disre- 
garded, and  the  certainty  insured  against  the  commission 
of  any  palpable  error  in  the  calculation.  The  friction  of 
the  knife  blades,  which  only  roll  upon  their  cushions, 


MOTION   OF  ROTATION. 


231 


may  be  neglected.  We  know,  moreover,  and  can  de- 
termine at  the  start,  the  position  and  the  traces  of  the 
vertical  plane,  which 
contains  the  centre  of 
gravity,  when  the  ap- 
paratus is  free.  It  is 
easy,  then,  to  measure 
the  inclination  which 
this  plane  takes  under 
the  action  of  a  given 
counterpoise.  This 
done,  call 

p  the  weight  of  the 
pendulum ; 

d  the  distance  sought 
of  its  centre  of  gravity 
from  the  axis  of  the 
knife  blades ; 

a  the    inclination 
of   the    plane    which 
passes  through  the  centre  of  gravity,  and  through  the  axis 
of  the  blades,  with  the  vertical ; 

L  the  perpendicular  let  fall  from  this  axis  upon  the 
direction  of  the  cord  ; 

T  the  tension  of  this  cord,  and  we  have  the  relation 


whence 


t=p .  d  sin  a ; 

£-&; 

^ .  sin  a 


195.  Centre  of  percussion. — When  a  body  (Fig.  95)  re- 
ceives a  motion  of  rotation  around  the  axis  A,  which  we 
suppose  here  as  perpendicular  to  the  plane  of  the  table, 
each  elementary  muss  of  this  body  develops  partial  forces 
of  inertia,  perpendicular  to  the  respective  distances  of 


232 


MOTION   OF  EOTATION. 


each  of  them  from  the  axis  whose  intensity  is  measured 

by  rar-,  according  to  the  notation  adopted  in  No.  186. 
t 

The  moment  of  each  of 
these  forces  in  respect 
to  the  axis  of  rotation  is 

mr* .  - -,  and  the  sum  of 

0 

all  the  similar  moments 

/y 

has  for  its  value  I .  -1. 
t 

If  we  decompose  each 

partial  force  m .  r-±  into 
t 

two  others,  the  one  hori- 
zontal and  the  other 
vertical,  and  call  x  and 
y  the  abscissa  and  ordi- 
nate  of  m  in  relation  to 
a  vertical  plane  and  a 
horizontal  plane  passing 
through  the  axis  A,  the  first  component  will  evidently  be 


FlG- 95- 


and  the  second  will  be 


v.  y    mv, 


v,  x  mv, 
->  -=  —  l 
t  T  t 


If  we  call  a?!  and  y1  the  co-ordinates  of  the  centre  of 
gravity  of  the  body,  we  shall  have,  in  making  separately 
the  sum  of  all  the  horizontal  and  vertical  components, 
according  to  the  theory  of  parallel  forces, 


MOTION  OF  ROTATION.  233 


and  [mse+m'a/+  .  .  .  ]= 

t  t 

whence  it  follows  that  the  resultant  of  these  two  groups 
of  rectangular  forces  is 


and  that  it  makes  with  the  horizontal  axis  and  the  verti- 
cal axis  of  co-ordinates,  angles  whose  cosines  are  respect- 

*?/  *¥ 

ively  ~  and  -^,  so  that  it  is  perpendicular  to  the  distance 
du          cL 

d  of  the  centre  of  gravity  from  the  axis. 

This  granted,  if  we  designate  by  O  the  point  of  appli- 
cation of  this  resultant  F=— M  .  d,  its  moment  will  be 

t 

equal  to  the  sum  of  those  of  all  the  forces  of  inertia  of  the 
body,  and  we  shall  have 


T' 

whence 

A0=~ 

M..d 

The  point  thus  determined  is  called  the  centre  of  per 
cussion.  It  is  such  that  a  force  capable  of  producing  in 
an  element  of  time,  the  variation  of  angular  velocity  vl9 
and  which,  when  applied  to  this  point,  shall  be  precisely 
equal  to  the  resultant  of  all  the  forces  of  inertia,  of  the 
different  elements  of  the  body.  Thus  the  pressure,  or,  as 
we  commonly  say,  the  percussion  upon  the  axis,  of  this 
force  and  this  resultant,  being  equal  and  directly  opposite, 
will  be  zero. 

Then,  reciprocally,  that  this  pressure  may  be  zero,  the 
exterior  force  producing  the  variation  of  motion,  must 
pass  through  the  centre  of  percussion,  so  that  no  shock 


234 


MOTION  OF  ROTATION. 


may  occur  upon  the  knife  blades  or  upon  the  axis  of 
rotation. 

We  would  observe  that  the  distance  of  the  centre  of 
percussion  from  the  axis  is  the  same  as  that  of  the  centre 
of  oscillation,  and  that  these  two  points  merge  into  each 
other.  This  is  the  reason  that,  in  ballistic  pendulums, 
they  are  so  arranged  as  to  receive  the  action  of  the  pow- 
der, or  the  shock  of  the  projectile,  precisely  at  the  height 
of  the  centre  of  oscillation. 

196.  Theory  of  the  ballistic  pendulum. — In  the  recep- 


FIG.  96. 


MOTION  OF  ROTATION.  235 

tion  and  testing  of  powder,  we  generally  make  us  of  a 
contrivance  known  by  the  name  of  ballistic  pendulum, 
(Fig.  96,)  the  inventor  of  which  was  Kobins,  a  celebrated 
English  Professor  of  Artillery,  and  which  has  lately,-  in 
France,  received  material  improvement. 

The  ballistic  pendulums  used  in  the  French  powder 
magazines,  whether  for  trials  of  guns  or  cannons,  are  com- 
posed of  a  cast  iron  receiver,  suspended  in  an  iron  frame. 
This  receiver  contains  soft  or  compressible  matter,  capa- 
ble of  receiving  and  deadening  the  shock  and  the  velocity 
of  a  projectile  without  any  rupture  of  the  receiver. 

The  firing  takes  place  at  the  height  of  the  axis  of  the 
receiver,  which  is  horizontal.  We  will  here,  as  in  the 
"  Aide-Memoire  des  officiers  d'artillerie,"  call 

B.  the  radius  of  the  arc  described  by  the  index  along 
the  graduated  limb,  showing  the  angles  of  the  recoil  : 

i  the  distance  of  the  point  shocked,  or  point  of  impact 
from  the  horizontal  plane  of  the  knife  blades  ; 

k  the  distance  of  the  centre  of  oscillation  from  the 
horizontal  plane  of  the  knife  blades  ; 

p  the  total  weight  of  the  loaded  pendulum,  that  is  to 
say,  including  the  buffers  pr  barrels  full  of  sand,  for  can- 
nons, or  the  block  of  lead  or  wood  for  guns  ; 

d  the  distance  of  the  centre  of  gravity  of  the  loaded 
pendulums  from  the  line  of  the  blades  ; 

5  the  weight  of  the  projectile  ; 

c  the  chord  of  the  arc  of  recoil  ; 


Y  the  velocity  of  the  projectile  at  the  instant  of  con- 
tact with  the  receiver  ; 

Y!  the  angular  velocity  imparted  to  the  pendulum 
after  the  shock. 

We  must  first  remark  that  during  the  shock  there  is 
developed,  at  the  point  of  contact  of  the  projectiles  and 
receiver,  efforts  of  action  and  reaction,  equal  and  directly 
opposite. 


236  MOTION  OF  EOTATION. 

The  action  exerted  upon  this  receiver  accelerates  its 
motion,  and,  from  what  precedes,  the  moment  of  this  force 
in  relation  to  the  axis  of  rotation  should  be  equal  to  that 
of  all  the  forces  of  inertia  of  the  material  molecules  com- 
posing the  pendulum. 

In  continuing  to  call  v1  the  small  increase  of  angular 
velocity  imparted  to  the  pendulum  during  the  element  of 
time  t)  the  resistance  of  an  elementary  mass  m,  situated 
at  the  distance  r  from  the  axis  will  be  expressed  by 

mr.  —  ]  its  moment  in  relation  to  the  axis  will  be  wr2.—  '; 
t  t 

AJ 

the  sum  of  all  the  similar  moments  will  be  I  .  -1,  and  should 

t  •", 

be  equal  to  the  moment  of  the  effort  exerted  at  the  same 
instant  by  the  projectile. 

But,  on  the  other  hand,  the  projectile,  acting  perpen- 
dicularly at  its  distance  i  from  the  horizontal  planes  of  the 
blades,  loses  in  an  element  of  time  a  small  degree  of 
velocity  v,  and  its  inertia,  which  is  the  same  for  all  the 
points  which  are  impressed  with  velocities  very  nearly 
equal  and  parallel,  occasions  a  motive  effort  expressed  by 

-.  -,  the  moment  of  which  in  relation  to  the  axis  of  the 

ff  * 

,  ,   ,     .    ~b  .  v 

blades  is  -•&.-. 

g   t 

Thus,  at  any  instant  of  the  shock,  we  must  have,  be- 
tween the  actions  developed  by  the  projectile  and  the 
reaction  of  the  pendulum,  the  relation 


or 

I.        T 
-^.v=Ivl. 

9 

In  establishing  analogous  relations  between  all  the  ele- 
mentary degrees  of  velocities,  lost  successively  by  the 


MOTION   OF   KOTATIOST.  237 

projectile  and  gained  by  the  pendulum,  we  shall  have,  in 
adding  them, 


Now,  the  sum  v+v'+v"  +  &c.,  is  evidently  equal  to  the 
total  velocity  lost  by  the  projectile,  from  the  moment  it 
struck  the  receiver  with  the  velocity  Y,  to  that  when, 
having  lost  all  relative  velocity  in  respect  to  the  receiver, 
it  partook  of  a  motion  in  common  with  it  equal  to  Y^',  in 
calling  Y!  the  angular  velocity  imparted  to  this  body  ; 
we  have  then 


On  the  other  hand,  the  receiver  starting  from  repose, 
and  acquiring  by  the  shock  the  final  angular  velocity  Y1? 
we  have 


The  above  relation  becomes  then 

v 


y  j 


snce    =*- 
9 
"We  deduce  from  this  expression 


w&Sf 

and  we  have  elsewhere  seen  that  we  must  have  i=k  in 
order  that  no  shock  should  be  produced. 

But  on  the  other  hand,  when  the  pendulum  recoils, 
its  centre  of  gravity  is  raised,  and  its  vis  viva,  as  well 
as  that  received  by  the  projectile,  being  soon  extin- 
guished, should  be  equal  to  double  the  work  developed 


238  MOTION   OF  KOTATION. 

by  gravity  and  by  the  friction  of  the  rolling  of  the  blades, 
which  we  have  neglected. 

The  angle  described  by  the  pendulum  being  a,  it  is 
clear  that  its  centre  of  gravity  is  raised  by  the  quantity 


d—  <#cos  a=d  (1  —  cos  a)  = 

The  projectile  was  at  rest  at  the  distance  i  from  the 
axis  of  rotation,  and  has  been  raised  the  height 


^ — ^  cos  a= 

v   ' 

then  the  work  developed  by  gravity  upon  the  pendulum 
and  the  ball  has  for  expression 

(pd+li)  2  sin2-#. 

The  vis  viva  possessed  by  these  two  bodies  at  the  end 
of  the  shock,  or  of  their  reciprocal  reaction,  is 

\7 
V 
We  have  then 

L* 

9 

whence 

-rr  /( Vd+bi)  Q     c.     . 

V  =  A  /  -^- — ! — £2. .  2  sin   , 
2 


Making  the  value  of  'Vl  equal  to  the  preceding  we  have 

Kir 


from  which  we  deduce 


li  2 


MOTION   OF  ROTATION.  239 

Such  is  the  formula  which  serves  to  calculate  the  initial 
velocities  of  projectiles  by  means  of  the  data  within  it  and 
of  the  angle  of  recoil. 

We  would  remark,  that  in  calling  C  the  chord  of  the 
arcs  of  recoil,  whose  radius  is  R,  we  have 

2sin-a-- 
which  gives 


y=  V(pdk+W)(pd+li)g  C 
U  '  R' 

This  is  the  form  given  in  "  Aide-Memoire  d'artillerie" 
We  have  seen  that  the  conditions  of  having  no  shock 
upon  the  blades  led  us  to  that  of  i=Jc.    If  it  was  com- 
pletely satisfied,  the  above  formula  would  be  reduced  to 


which  shows  that  then  the  measured  velocities  would  be 
proportional  to  the  chords  of  the  arcs  of  recoil. 

But  this  condition  which  has  been  nearly  attained  in 
the  construction  of  the  new  pendulums  for  cannons  at 
Metz,  and  Yincennes,  and  of  Bouchet,  is  by  no  means 
satisfied  in  the  pendulums  for  guns,  and  this  accounts  for 
their  so  sensible  vibrations. 

Officers  of  artillery  will  find  farther  details  upon  these 
contrivances  in  the  instructions  for  semi-annual  trials  of 
powder  with  ballistic  pendulums. 


GENEEAL  APPLICATION  OF  THE  PKESTCIPLE 
OF  VIS  VIVA  TO  MACHINES. 

197. — Application  of  the  principle  of  vis  viva  to  ma- 
chines.— In  applying  this  principle  to  the  motion  of  ma- 
chines, we  must  examine  separately  the  circumstances 
and  conditions  of  action  of  the  different  forces  to  which 
they  are  subjected.  These  forces  may  be  classified  as 
follows : 

1st.  The  powers  which  produce,  maintain,  or  acceler- 
ate motion,  and  whose  work,  which  we  shall  designate  by 
F .  S,  is  always  developed  in  the  direction  of  the  motion, 
and  is  consequently  positive;  we  designate  by  F  the 
mean  effort  of  the  resultant. 

2d.  The  useful  resistances  which  must  be  overcome  or 
destroyed  to  produce  the  effect  proposed  or  the  work 
which  the  machine  is  to  do,  and  which  destroy,  retard,  or 
moderate  the  motion.  The  work  of  these  resistances, 
which  we  shall  designate  by  Q  .  S',  is  always  developed 
in  an  opposite  direction  to  that  of  the  powers  and  must 
be  subtracted. 

3d.  The  prejudicial  or  passive  resistances,  existing 
in  motion  such  as  frictions,  the  resistance  to  rolling, 
that  of  the  air,  of  water,  &c.,  which  absorb  unprofitably 
a  portion  of  the  motive  work,  and  retard,  moderate,  or 
destroy  the  motion,  and  whose  work  we  shall  represent 
by  E  .  S",  is  always  to  be  subtracted  from  that  of  the 
powers. 

4th.  The  action  of  gravity,  which  should  be  regarded 


PRINCIPLE   OF   VIS    VIVA   APPLIED   TO   MACHINES.          24:1 

separately  whenever  it  acts,  sometimes  as  a  power,  some- 
times a  resistance,  and  its  work,  represented  by  P  .  H, 
will  be  positive  or  additive  in  regard  to  that  of  the  pow- 
ers in  the  first  case,  and  negative  or  subtractive  in  the 
second.  But,  when  gravity  acts  always  as  a  power,  as  in 
hydraulic  wheels,  clock-weights,  &c.,  it  should  be  reck- 
oned among  the  powers  ;  and  inversely  when  it  acts  as  a 
useful  resistance,  as  in  machines  for  raising  weights,  &c., 
it  should  be  joined  with  the  useful  resistances. 

With  this  classification  of  forces,  the  principle  of  vis 
viva  will  be  represented  by  the  equation 


[V/a-  Y,2]  :=FS-QS'-K  .  S"±'PH, 

a 

in  case  the  machinery  is  composed  of  rotating  pieces  ;  or 
in  general 

•hi  [V—  V]=FS-QS'-K  .  S"±PH  ; 


the  expressions  IY/2,  MY/a,  etc.,  representing  the  sum  of 
all  the  analogous  vis  vivas  of  the  parts  of  the  machine. 

This  relation  refers  to  a  finite  interval  of  time,  and  we 
have  seen  that  for  an  element  of  time,  or  an  infinitely 
small  displacement,  we  have  also 


The  aim  in  the  establishment  of  every  machine,  being  to 
overcome  a  useful  resistance,  or  to  do  a  certain  work,  it 
is  evident  that  it  is  the  work  QS'  or  Qs'  of  these  useful 
resistances,  which  should  be  rendered  the  greatest  possi- 
ble, or  a  maximum  ;  if  we  deduce  from  the  above  relation 
the  value  of  QS'  we  have 


16 


242         PRINCIPLE   OF  VIS  VIVA   APPLIED   TO   MACHINES. 

or  for  the  element  of  time 


198.  Conditions  of  the  'maximum  of  effect  of  ma- 
chines. —  Let  us  examine  successively  the  conditions  which 
should  be  satisfied,  for  a  maximum  of  useful  work. 

199.  Work  of  powers,—  W&  remark,  first,  that  for  each 
kind  of  motor  or  of  power,  there  is  a  maximum  effort  cor- 
responding to  a  velocity  zero,  for  which  the  work  is  zero, 
and  a  maximum  velocity  corresponding  to  an  effort  zero, 
where  also  the  work  is  nothing.     Thus  for  animal  motors 
the  effort  and  the  velocity  have  absolute  limits,  for  which 
the  one  is  zero,  when  the  other  is  a  maximum.     It  is  the 
same  for  hydraulic  wheels,  for  which  the  effort  is  a  maxi- 
mum when  the  velocity  is  zero,  and  at  its  minimum  when 
the  velocity  is  the  greatest  which  the  water  can  impart  to 
its  course  in  open  space.     It  is  also  the  same  for  steam 
machines,  wind-mills,  &c. 

Between  these  limits  there  is  a  certain  velocity  which, 
for  each  motive  power,  according  to  its  nature  and  its 
combination  of  mechanical  parts,  corresponds  to  a  maxi- 
mum quantity  of  work,  developed  by  the  power,  and  as 
it  often  happens  that,  for  the  greatest  or  smallest  veloci- 
ties, the  work  diminishes  rapidly,  it  follows  that  it  is  very 
important  to  preserve,  at  the  points  of  application  of  the 
motive  power,  the  velocity  which  corresponds  to  its  maxi- 
mum of  effect,  and  therefore  a  uniform  motion  for  the 
recipient  of  the  power. 

200.  Work,  of  useful  resistances.  —  We  make  the  same 
observations  for  the  work  of  useful  resistances,  for,  accord- 
ing to  the  nature  of  the  tools  and  the  products,  there  is  a 
certain  velocity  which  answers  to  the  best  quality  of  pro- 
ducts, the  best  effect,  or  the  longest  duration  of  the  tools  ; 
thus  for  the  grinding  of  corn,  the  rolling  of  iron,  for  the 


PRINCIPLE    OF   VIS    VIVA   APPLIED   TO   MACHINES.         243 

drawing  and  spinning  of  cotton,  of  wool,  &c.,  there  is  a 
velocity  suited  to  the  quality  and  nature  of  the  products 
to  be  obtained ;  in  saw-mills,  the  turning  of  metals, 
pumps,  &c.,  the  preservation  of  the  tools,  or  the  economy 
of  work,  exacts  a  velocity  within  certain  limits,  &c.  Then, 
also,  it  is  best  that  there  should  be  a  uniform  motion  for 
useful  resistances,  as  well  as  for  motive  powers. 

201.  The  work  of  prejudicial  or  passive  resistances. — 
As  to  prejudicial  or  passive  resistances,  the  work  which 
they  consume  being  always  expended  at  a  loss,  we  must 
evidently  seek  to  render  them  the  smallest  possible.     It 
is  necessary,  then,  to  diminish  the  friction,  and  conse- 
quently the  weight  of  the  pieces  which  slide  upon  each 
other ;  to  polish  their  surfaces,  to  keep  them  well  oiled, 
and  to  diminish  the  spaces  described  by  the  rubbing  parts. 
For  the  resistance  of  the  air  or  of  water,  we  should  limit 
the    velocities,  and  give  to  the  bodies  the  forms   best 
adapted  to  lessen  these  resistances,  etc. 

202.  Pieces  with  alternating  motion. — The  work  due 
to  the  weight  of  pieces  alternately  and  periodically  as- 
cending and  descending  the  same  height,  being  zero  for 
each  period,  we  see  that  there  is  no  occasion  to  concern 
ourselves  with  machines  whose  motion  embraces  a  great 
number  of  similar  periods,  if  these  alternatings,  while  in- 
creasing or  diminishing  periodically  the  motive  work,  do 
not  produce  corresponding  variations  in  the  motion,  and 
so  alter  the  uniformity  of  motion,  the  necessity  of  which 
has  been  recognized. 

If,  then,  we  cannot  wholly  suppress  the  pieces  which 
ascend  or  descend  periodically,  it  would  be  well  to  limit 
their  number  and  influence  as  much  as  possible,  and  the 
general  condition  will  be  to  employ  only  pieces  well  cen- 
tred in  relation  to  their  axis  of  rotation,  or  whose  centre 
of  gravity  remains  at  the  same  height. 

On  this  subject,  we  could  show  a  dynamometric  ex- 


244        PRINCIPLE   OF  VIS   VIVA   APPLIED   TO   MACHINES. 

perimental  curve,  obtained  upon  a  ventilator  which,  by 
the  nature  of  the  resistance  to  be  overcome,  should  have 
been  a  uniform  motion,  but  which,  by  reason  of  a  defect 
in  centring,  presented,  on  the  contrary,  very  considera- 
ble periodical  variations. 

"What  we  have  said  of  pieces  which  rise  and  fall  peri- 
odically under  the  action  of  gravity  refers  also  to  pieces 
with  alternating  motion,  such  as  the  horizontal  frames  of 
saws,  etc.,  whose  variable  vis  viva  is  opposed  to  the  uni- 
formity of  motion. 

203.  Influence  of  the  vis  viva  possessed  or  acquired  at 
each  period. — We  see  by  the  equation  of  the  principle  of 
vis  viva,  that  if  the  vis  viva  has  diminished  during  the 
period  considered,  the  half  of  this  diminution  represents  a 
work,  which  is  added  to  that  of  the  motor,  and  that  if,  on 
the  contrary,  the  vis  viva  has  increased,  the  half  of  its 
variation  represents  the  portion  of  the  motive  work  ab- 
sorbed to  produce  it.  If,  then,  the  motion  is  periodical, 
we  see  that  in  the  accelerations  of  motion,  the  inertia  of 
the  masses  absorbs  and  stores  up  a  portion  of  the  motive 
work,  which  it  restores  in  its  retardations.  Inertia,  then, 
performs  here  truly  the  duty  of  a  reservoir  of  work,  abso- 
lutely like  the  pond,  the  reservoir  of  the  hydraulic  wheel, 
which  receives  and  preserves  the  water  of  a  stream,  when 
the  wheel  does  not  consume  all  the  flowing  water,  and,  on 
the  other  hand,  furnishes  it,  in  emptying  the  water  con- 
sumed by  the  wheel,  when  this  wheel  expends  more  than 
the  supply  of  the  source. 

Examples  of  these  effects  are  as  numerous  as  remarka- 
ble in  the  working  of  machines :  thus,  in  the  working  of 
rolling-mills,  which  are  set  in  motion  before  passing  the 
iron  between  the  cylinders,  all  the  pieces  of  the  machine 
receive  an  accelerated  motion,  and  absorb  a  considerable 
portion  of  the  motive  work ;  thus,  when  we  pass  the  metal 
to  be  drawn,  the  work  of  resistance  prevails  over  that  of 


PBINCIPLE   OF   VIS   VIVA  APPLIED   TO   MACHINES.        245 

the  power,  the  motion  is  retarded,  and  the  inertia  of  the 
masses  restored,  develops  in  favor  of  the  motor  the  quan- 
tity of  work  which  it  had  previously  absorbed.  It  is  the 
same  in  the  action  of  walking-beams,  of  hammers,  of  the 
treadles  of  knife-grinders,  &c. 

But  as  these  variations  of  vis  viva  correspond  with 
variations  of  velocity,  it  becomes  us  to  restrain  them  and 
to  limit  them  as  far  as  possible,  so  as  to  obtain  as  near  an 
approach  to  uniformity  as  may  be. 

204.  Case  of  periodical  motion. — There  are  many  ma- 
chines which,  by  their  constitution,,  or  by  the  nature  of 
their  work,  cannot  be  impressed  with  uniform  motion. 
Of  this  number  are  all  those  where  the  motors  or  the  tools 
act  intermittently  or  in  alternate  directions,  as  steam  en- 
gines, or  a  column  of  water  on  one  side,  and  on  the  other 
saws,  pumps,  hammers,  &c.     In  all  such  cases  it  is  neces- 
sary to  reduce   the   number  of   pieces  impressed  with 
alternating  motion  to  what  is  strictly  necessary,  and  to 
distribute  the  variations  of  resistance  or  of  work  among 
equal  spaces. 

When,  by  these  means,  we  have  attained  an  exactly 
periodical  motion,  and  when  the  vis  viva  absorbed  in  the 
accelerations  is  restored  in  the  retardations,  we  may,  in 
calculating  the  effect  of  an  entire  period,  dispense  with 
the  reckoning  of  the  vis  viva,  which  will  be  zero  for  the 
entire  duration  of  this  period. 

205.  Advantages  and  conditions  of  uniform  'motion. — 
But,  in  general,  uniform  motion  being  the  most  favorable 
to  the  action  of  motors  and  of  tools,  and  occasioning  less 
loss  of  work  by  the  effect  of  the  passive  resistance,  since 
it  admits  of  giving  to  all  the  pieces  of  machines  smaller 
dimensions,  and  accordingly  less  weight,  it  follows  that 
we  should  try  all  possible  means  to  obtain  it,  or  at  least 
to  approximate  to  it. 


24:6         PRINCIPLE    OF   VIS    VIVA   APPLIED   TO   MACHINES. 

It  becomes  us,  then,  to  use,  if  possible,  as  organs  for 
the  transmission  of  motion,  parts  with  a  continuous  mo- 
tion, with  their  centres  of  gravity  resting  at  the  same 
height,  with  wheels  exactly  centred,  &c.,  to  distribute 
the  materials  to  be  worked  in  a  continuous  manner,  or  at 
least  at  equal  intervals,  as  is  done  with  the  "babillard 
des  moulins  "  of  mills,  the  claws  of  saw-mills,  etc. 

206.  Inconvenience  of  variable  motion  and  means  of 
diminishing  it.  —  Besides  the  inconveniences  which  we 
have  pointed  out,  relative  to  the  irregular  action  of  the 
motive  power  and  of  the  useful  resistance,  there  is  another 
which  obliges  us  to  give  to  the  parts  subjected  to  it  larger 
dimensions  than  those  required  by  uniform  motion  for  the 
same  work,  since  the  efforts  which  these  pieces  have  to 
resist  are,  at  certain  instants,  much  greater  than  the  con- 
stant effort  corresponding  to  uniform  motion.     From  this 
results  an  excess  of  weight  and  an  increase  of  friction,  be- 
sides the  shocks  or  the  more  or  less  sensible  alterations 
of  forms  produced  by  changes  of  velocity. 

All  these  inconveniences  being  greater  as  the  vis  viva 
of  the  pieces  with  alternating  motion  are  more  considera- 
ble, it  will  be  requisite,  after  having  limited  their  dimen- 
sions to  what  may  be  necessary,  to  make  their  velocities 
as  small  as  possible  in  relation  to  those  of  the  pieces  en- 
dowed with  a  uniform  motion,  or  one  approximating  to  it. 

207.  Observations  upon  the  starting  of  machines  and 
the  variations  in  velocity  which  then  take  place.  —  The 
relation 


gives  us  for  the  elementary  variation  of  velocity 


PKINCIPLE   OF   VIS    VIVA   APPLIED   TO   MACHINES.        247 

We  see  that  the  velocity  will  increase  when  the  elemen- 
tary motive  work  Fs  is  greater  than  the  sum  of  the  quan- 
tities of  work  of  all  the  resistances  ;  but  that,  for  a  given 
excess  of  work,  the  variation,  or  the  increase  of  velocity, 
will  be  so  much  the  less  as  the  velocity  Va  possessed  by 
the  body  is  greater,  and  as  the  moment  of  inertia  I  of 
masses  in  motion  is  the  more  considerable.  Also,  when 
the  elementary  motive  work  is  inferior  to  the  work  of  the 
resistances,  the  velocity  decreases,  but  so  much  the  less 
as  the  velocity  and  the  moment  of  inertia  are  the  greater. 

Rapid  motions,  and  those  in  which  the  moments  of 
inertia  are  considerable,  are  then  the  more  stable  ,  and  ex- 
perience less  alteration  from  the  action  of  given  causes. 

When  a  machine  starts  from  rest,  its  velocity,  at  first 
zero,  increases  gradually,  since  the  work  of  the  motor 
prevails  at  each  instant  over  that  of  the  resistance.  But, 
on  the  one  hand,  the  motive  work  attains  its  maximum 
value  at  a  certain  velocity,  having  passed  which  it  de- 
creases ;  and,  on  the  other  hand,  the  work  of  the  resist- 
ance increases  often  with  the  velocity,  so  that  soon  we 
have  the  equality 

' 


At  this  instant  the  variation  or  the  increase  i>r  of  the 
velocity  is  zero,  and  the  velocity  has  attained  its  maxi- 
mum. If  this  equality  of  motive  work  and  of  resistant 
work  subsists,  the  nlotion  becomes  uniform  ;  but  this  can- 
not happen  except  the  term  =pPA  is  zero  ;  that  is  to  say, 
that  the  centre  of  gravity  of  all  the  pieces  remains  always 
at  the  same  height. 

This  condition  of  uniform  motion  is  in  some  sort  self- 
evident,  since  it  amounts  to  saying  that  the  work  of  pow- 
ers tending  to  accelerate  or  maintain  motion  should  be 
equal  to  that  of  the  resistances  which  tend  to  retard  or 
destroy  it. 

The  elementary  work  being,  as  we  have  already  re- 
marked in  No.  120,  what  is  termed,  in  rational  mechanics, 


24:8         PKINCTPLE   OF   VIS   VIVA   APPLIED   TO   MACHINES. 

the  virtual  moment,  we  see  that  the  preceding  statement 
amounts  to  saying  that,  for  uniform  motion,  or  for  equi- 
librium, which  is  but  a  particular  case  of  it,  the  virtual 
moment  of  powers  must  be  equal  to  that  of  resistances,  or 
their  sum  equal  to  zero. 

208.  Observation  relative  to  perpetual  motion.  —  The 
velocity  only  remaining  the  same  when  the  elementary 
variation  ^=0,  we  should  then  have 


ISTow,  in  supposing,  even,  the  work  of  useful  resistance 
Qsf  to  be  zero,  in  which  case  the  machine  serves  no  useful 
purpose,  that  of  the  prejudicial  resistances  Ks"  can  never 
be  zero,  since  we  cannot  have  machines  without  weight, 
and  consequently  without  friction.  "We  must  then  always 
have  a  certain  motive  work  Fs  to  maintain  the  motion, 
which  shows  the  absurdity  of  all  the  attempts  to  obtain 
what  is  called  perpetual  motion,  or  a  motion  self  sus- 
taining, without  the  aid  of  any  exterior  motive  force. 

209.  Periodical  motion.  —  It  seldom  happens  that  the 
motive  work  remains  always  equal  to  that  of  the  resist- 
ances starting  from  the  instant  when  the  velocity  has  ac- 
quired its  maximum  value  ;  most  usually,  on  the  contrary, 
the  resistant  work  begins  then  to  prevail  over  the  motive 
work,  the  variation  in  the  velocity  becomes  negative,  and 
the  motion  slackens.  But  as  the  work  of  useful  or  passive 
resistances  may  diminish,  while,  at  the  same  time,  that 
of  the  power  increases,  the  excess  of  the  first  above  the 
second  diminishes,  the  motion  is  retarded  less  and  less, 
and  we  have  again 


The  velocity  ceasing  then  to  diminish,  it  attains  its  mini- 
mum. 

If  the  diminution  of  the  velocity  does  not  cancel  the 


PRINCIPLE   OF   VIS   VIVA   APPLIED   TO   MACHINES.        249 

motion,  there  follows  then  another  period  of  acceleration, 
limited  by  a  second  maximum,  and  so  on. 

Machines  then  work,  for  the  most  of  the  time,  with  a 
periodical  motion,  sometimes  accelerated,  sometimes  re- 
tarded, in  which  the  velocity  attains  successively  and 
alternately  maxima  and  minima*;  but,-these  periods  being 
accomplished  usually  in  equal  times,  we  substitute,  as  we 
have  said,  for  this  variable  velocity,  quite  difficult  to  be 
determined,  the  consideration  of  a  mean  velocity. 

210.  Manner  of  limiting  the  deviation  of  velocity  — 
Theory  of  fly-wheels.  —  After  having  used  all  ordinary 
means  to  regulate  the  play  of  machines,  there  remains 
still  another  to  restrain  the  variations  of  the  velocity,  be- 
tween suitable  limits  for  each  case,  under  the  action  of 
given  and  alternating  excesses  of  the  motive  or  resistant 
work. 

In  fact,  if  we  consider  the  equation  by  means  of  which 
we  express  the  principle  of  vis  viva. 

I  [Y/2—  Y,2]  =2  [FS-QS/-ES//±PH]=2W, 

we  see  that,  for  a  determinate  period,  in  which  the 
velocity  shall  have  varied  from  Yx  to  Y/,  under  the  influ- 
ence of  a  given  excess  "W  of  motive  work  above  the 
resultant  work,  the  variation  of  the  squares  of  the  veloci- 

ties 


V/2          V3  — 
V   1       "         V    1    ~~  •"r"" 

will  be  so  much  the  smaller  as  the  moment  of  inertia  of 
the  pieces  endowed  with  the  motion  of  rotation,  or  the 
mass  of  the  pieces  impressed  with  the  motion  of  transla- 
tion, are  more  considerable.  Thus,  after  having  by  a 
good  disposition  of  machines,  by  a  symmetrical  distribu- 
tion of  the  resistances,  etc.,  diminished,  as  far  as  possible, 
the  alternating  excess  of  work  causing  the  irregularity, 


250        PRINCIPLE   OF   VIS   VIVA   APPLIED   TO   MACHINES. 


we  may  check,  as  far  as  we  wish,  the  variation  of  velocity, 
by  increasing  the  moment  of  inertia  or  the  mass  of  the 
movable  pieces,  or  more  simply,  the  moment  of  inertia 
of  one  of  them  specially  appointed  for  this  end. 

This  piece  is  called  the  fly-wheel,  and  is  usually  com- 
posed of  a  cast-iron  ring  of  great  diameter,  with  cast-iron 
arms,  and  is  placed  as  near  as  possible  to  the  parts  of  the 
machine  impressed  with  variable  motion,  in  order  that 
their  irregularities  may  be  lessened  in  the  transmission  to 
the  other  parts. 

In  the  establishment  of  the  fly-wheel,  we  usually 
neglect  the  regulating  influence  of  the  other  masses  which 
nevertheless  contribute  towards  insuring  a  greater  regu- 
larity than  could  be  attained  by  the  fly-wheel  alone. 

We  would  h'rst  remark,  that  the  difference  of  the 
squares  of  the  velocities, 


which  is  evident  by  an  examination  of  the  figure,  where 

AB=Y/  and  EF=Y,. 
In  fact,  we  see  that 


=AIHK=AIxIH=(Y1/+Y1)  (VY—  V,). 


E\ 


If,  further,  we  call  U 
the  arithmetical  mean 


FIG.  97. 


•jffOO         -3- 

between  the  velocities 
Y/  and  Y15  we  remark 
that  U  will  differ  but 
very  little  from  the  mean 
velocity  of  the  machine 


PRINCIPLE   OF   VIS   VIVA  APPLIED  TO   MACHINES.        251 

derived  from  the  number  of  turns  which  it  makes,  and 
which  is  usually  given  beforehand,  according  to  the  ar- 
rangement of  the  machine  ;  we  shall  have  then 


and  consequently 
whence 


If  now,  to  obtain  a  given  degree  of  regularity,  we  im- 
pose the  condition  that  the  angular  velocity  shall  not  vary 

over  a  fraction  -  of  the  mean  velocity  IT,  we  shall  have 


and  consequently 

U_W 

7*~UI; 

from  which  we  deduce 

nW 


1= 


IP 


We  see,  then,  that  when  the  excess  of  the  motive  work 
above  the  resistant  work,  or  vice  versa,  is  given,  as  well 
as  the  mean  angular  velocity  of  rotation  of  the  shaft  of 
the  fly-wheel,  and  the  regulator  number  n,  we  may  de- 
duce from  this  simple  expression  the  moment  of  inertia 
of  the  fly-wheel. 

We  observe  that  the  moment  of  inertia  will  be  so 
much  the  smaller  as  the  mean  angular  velocity  is  the 
greater,  and  that  consequently  it  is  proper  to  place  the 
fly-wheel  upon  the  axle  whose  motion  is  the  most  rapid. 


252      PRINCIPLE   OF   VIS   VIVA  APPLIED   TO   MACHINES. 

In  calculating  the  moment  of  inertia  of  a  fly-wheel, 
we  usually  neglect  the  influence  of  the  arms,  and  we 
have  then  very  nearly 

V 
I=—  E2, 

g 

P'  being  the  weight  of  the  ring,  and  R,  its  mean  radius. 
Moreover,  we  know  that 


a  and  5  being  the  width  and  thickness  of  the  ring,  R,  its 
mean  radius,  and  <fc455.25lbs-  the  weight  of  a  cubic  foot  of 
cast-iron.  If  we  wish  to  take  into  the  account  the  influ- 
ence of  the  arms,  the  moment  of  inertia  has  for  an  ap- 
proximate value 


9 

in  putting  P=P'+0.325P". 

As  for  the  regulating  number  n,  it  depends  upon  the 
nature  of  the  machine,  and  the  quality  of  the  products 
to  be  obtained,  and  cannot  always  be  the  same  for  any 
given  class  of  machines.  Thus  for  steam-engines,  the 
degree  of  regularity  depends  upon  the  products  to  be  ob- 
tained; and  if  for  many  cases  it  may  without  incon- 
venience be  the  same,  for  others  it  may  vary.  For  the 
spinning  of  cotton,  of  linen,  of  wool  ;  for  the  making  of 
paper  by  machinery,  this  number  should  be  increased 
according  to  the  perfection  of  the  products  to  be  fabri- 
cated. For  rolling-mills,  it  is  not  necessary,  as  is  too 
frequently  done,  to  adopt  the  same  fly-wheel  when  we 
are  to  draw  out  large  pieces  of  iron,  or  great  plates  of 
sheet-iron,  which  occasion  great  irregularities,  and  whose 
work  is  performed  at  intervals,  that  we  do  for  the  con- 
tinuous rolling  for  hours  of  small  plates  which  follow 
rapidly,  one  after  the  other.  It  is  by  observation  of  the 


PRINCIPLE    OF    VIS    VIVA   APPLIED   TO    MACHINES.          253 

action  of  good  machines,  and  by  calculation,  that  we  ar- 
rive at  the  determination  of  the  proper  degree  of  regu- 
larity for  each  special  case.  We  will  give  hereafter  the 
complete  theory  and  a  graphic  solution  of  the  question  of 
fly  wheels  of  stearn-engines ;  and  for  the  present  rest  con- 
tented, after  having  laid  down  the  fundamental  principles, 
with  giving  the  usual  practical  formulas  for  many  im- 
portant cases. 

211.  High-pressure  steain-engines. — In  this  case  we 
use  the  following  formulae,  according  as  the  length  of  the 
connecting-rods  is  equal  to 


6  times  the  crank  PV2=124092- 

m 

with 


PV2=131242 


m 


nN 
PY2==138392-— 


walking-beam. 


m    >  walking-beam. 

In  these  formulae,  N  is  the  nominal  force  in  horse-pow- 
ers, m  the  number  of  turns  of  the  fly-wheel  in  1",  Y  the 
mean  velocity  of  the  mean  circumference  of  the  ring. 

The  number  n,  according  to  the  common  practice  of 
Watt,  is  usually  equal  to  32,  for  all  cases  not  requiring 
extraordinary  regularity.  For  flour-mills,  saw-mills,  &c., 
it  may  be  diminished  a  little,  while  for  spinning  it  may 
be  increased  as  high  as  from  50  to  60. 

The  fly-wheel  for  the  spinning-mill  of  Logelbach 
affords  us  the  following  data  : 

Diameter  of  fly-wheel  20.01ft-     w=19, 

j  =35  horae  power. 

60 


254        PRINCIPLE   OF   VIS    VIVA    APPLIED    TO   MACHINES. 

If  we  make  7i=40, 


124091.5     40  x  35 
_ 


-D  . 

:_____     __  .23067  pounds. 


For  n=35,  we  have  P=20183  pounds.  The  con- 
structors have  set  P=20555  pounds. 

212.  flywheels  for  expansion  engines.  —  The  irregu- 
larity of  action  of  steam  being  very  great,  the  fly-wheel 
should  be  increased,  and  I  give  here,  for  examples,  for- 
mulae for  high-pressure  engines  with  expansion,  without 
condensation,  at  five  atmospheres  of  pressure  in  the  boiler. 

nN 
Expansion  commencing  at  |  the  stroke,  PY2=  168089  -  . 

PV2=193864—  . 
m 

PV2=218849 


in 

EXAMPLE.  —  Let  N=40  horse  powers,  expansion  com- 
mencing at  a  half  of  the  stroke, 

,=32,  m=16,  D=26.739«,  y^Ux  26. 

and  consequently 

168089x32x40 


213.  Fly-wheels  for  forge  hammers.  —  In  machines 
which  work  by  shocks,  such  as  trip-hammers,  the  irregu- 
larity of  motion  arises  from  the  intermittent  action  of  the 
resistance,  and  the  losses  of  work  produced  by  the  shock. 
"We  can  submit  these  effects  to  calculation,  to  determine 
directly  the  loss  of  vis  viva,  and  consequently  limit  the 
variations  of  the  velocity,  to  a  given  fraction  of  the  mean 


PRINCIPLE    OF   VIS    VIVA    APPLIED    TO    MACHINES-          255 

velocity ;  but  this  is  no  place  to  unfold  the  theory,  and 
we  confine  ourselves  to  saying  that  it  has  led  to  the  fol- 
lowing formulae : 

214.  "  Frontaux  "  or  Tennant  helves  hammers. 
From  6616  to  7718lb%  striking  70  to  80  blows, 

p_  474890 

From  8822  to  10S07lb%  striking  72  to  80  blows, 
712213 


The  regulating  number  has  been  taken  in  these  cir- 
cumstances equal  to  from  50  to  55  nearly. 

EXAMPLE  : 


The  "  frontal  "  hammer  of  the  forge  at  Framont, 
weighing  6616lbs-  and  upwards,  has  a  fly-wheel  with  ra- 
dius K=7.054ft-,  whose  ring  weighs  9327lbs-,  and  has 
worked  for  nearly  12  years. 

215.  German  hammers  geared, 

Weighing  from  1323  to  1764lbs>,  including  "  manche 
et  hurasse,"  beating  100  to  110  strokes  in  V. 


At  the  works  of  the  new  mill  connected  with  the  foun 
dries  at  Hyange, 

K=5.413ft  ;  P=11358lbs-  nearly. 


OF  THE 


256         PRINCIPLE    OF 'VIS    VIVA   APPLIED   TO    MACHINES. 

216.   Geared  tilt-hammers. 

Beating  from  150  to  200  blows  in  1',  794lbs-  in  weight, 
including  all, 

p=142442 
Ea 

Beating  from  150  to  200  blows  in  1',  and  weighing 
1103lb%  all  told, 

-p_213664 


217.    Vertical  saws  for  cutting  large  timber. — Obser- 
vations show  that  it  suffices  to  take 


p_712213 


EXAMPLE — Saw  mill  at  Mete. — The  radius  K=2.49f% 
the  number  of  turns  of  the  saw  is  88  in  1',  whence  we 
conclude 

CO 

V=|?6.28  x  2.49=22.93"-  »^ 

The  formula  gives 

p_  712213  lt, 

~(2O8)'~ 

We  usually  place  two  fly-wheels,  each  one-half  of  the 
above  weight.  At  the  saw  mills  of  Metz,  the  two  fly- 
wheels weighed  together  but  1129  pounds. 

218.  Necessity  of  using  fly-wheels  in  machines  when 
there  are  shocks. — A  striking  example  of  the  necessity  for 
using  fly-wheels  where  shocks  are  produced,  was  observed 
in  1845,  at  the  powder-mills  of  Youges,  and  at  that  of  St. 


PRINCIPLE   OF   VIS   VIVA   APPLIED   TO   MACHINES.        257 

Ponce,  in  four  stamping-mills.  In  substituting  for  these 
new  constructions  cast-iron  gearings,  in  place  of  the  old 
wooden  wheels,  care  had  been  taken  to  increase  in  the 
ratio  of  2  to  3,  the  dimensions  of  the  teeth  and  of  the 
wheels,  furnished  by  the  ordinary  rules  of  practice.  Not- 
withstanding this  precaution,  these  mills  having  been  set 
at  work,  the  wheels  of  the  gearing  could  not  resist  the 
vibrations  produced  by  the  shocks,  and  were  broken  at 
the  rims  after  a  short  service.  To  remedy  this  evil,  two 
methods  presented  themselves;  one,  which  consisted  in 
increasing  considerably  the  dimensions  of  the  wheels  was 
adopted  from  the  necessity  of  the  case,  for  the  broken 
wheels.  The  other,  the  most  rational,  was  to  place  fly- 
wheels upon  axles  with  cams,  to  diminish  the  vibrations 
of  the  velocity,  and  consequently  the  shocks  between  the 
wheels  and  pinions.  It  was  perfectly  successful,  and  the 
gearing  of  the  fourth  mill,  exactly  similar  to  those  which 
had  been  broken  when  there  was  no  fly-wheel,  had  resisted 
well,  with  the  employment  of  this  means  of  regulation. 

219.  Proportions  of  fly-wheels  for  powder-mills  with 
twenty  stamps. — The  stamps  of  powder-mills  weigh  from 
88  to  92.4  pounds,  and  beat  56  blows  per  minute,  there 
being  two  for  each  turn  of  the  shaft  with  cams.     Experi- 
ence has  proved  that  fly-wheels  of  8.2ft-  diameter,  .557ft- 
of  width  at  the  crown  in  the  direction  of  the  axle,  and 
,59ft-  in  that  of  the  radius,  were  sufficient. 

220.  Rolling-mill  for  great  plates  and  l)ulky  iron. — 
In  these  machines,  observation  shows  that  we  may  calcu- 
late the  fly-wheel  by  the  following  formula  : 

p_  3086258KK . 
wVa 

N  being  the  force  in  horse  powers  transmitted  to  the 
shaft  of  the  fly-wheels  ; 
17 


258        PRINCIPLE    OF   VIS    VIVA   APPLIED   TO   MACHINES. 

Y  the  mean  velocity  of  the  middle  circumference  of 
the  ring. 

m  the  number  of  turns  of  the  fly-wheel  (usually  placed 
upon  the  same  axis  as  the  cylinder),  in  V  ; 

K  a  constant  numerical  co-efficient  which  we  may 
take  equal  to ; 

K=20  for  machines  from  80  to  100  horse  powers,  and 
with  6  to  8  equipments  of  cylinders  ; 

K=25  for  machines  of  60  horse  powers,  and  from  4  to 
6  fixtures  of  cylinders  ; 

K=80  for  machines  from  30  to  40  horse  powers,  with 
one  or  two  cylinders  for  great  sheets  of  iron. 

EXAMPLE.— D =9.3177"-,  w=60,  V=60.368ft-  for  six 
cylinders  working  together, 

p_3086258  x  60  x 25^       »  lbs 
60  (60.368)2 

The  manufactory  at  Fourchambault,  placed  in  these 
circumstances,  has  a  fly-wheel  of  17643  pounds  only. 

When  the  machines  to  be  regulated  have  for  a  motor 
hydraulic  wheels  with  rapid  motion,  such  as  wheels  with 
plane  and  curved  floats,  the  moment  of  inertia  being 
usually  considerable,  it  may  be  added  to  that  of  the  fly- 
wheel, which  may  then  be  somewhat  diminished,  espe- 
cially if  the  motor  is  near  the  resistance. 

221.  Use  of  fly-wheels, — It  follows  from  what  has  been 
said,  that  fly-wheels  have  for  their  object  the  confining  of 
the  velocity  within  given  limits,  when  there  is  in  the 
course  of  th'e  pieces,  or  in  the  action  of  the  motors  or  of 
the  resistances,  inequalities  or  inevitable  alternations,  or, 
in  certain  cases,  to  accumulate,  during  a  portion  of  the 
periods  of  motion,  a  quantity  of  motive  work,  to  be  re- 
stored when  the  work  of  resistance  prevails  over  that  of 
the  motor.  It  is,  then,  only  momentarily  that  the  use  of 


PRINCIPLE   OF   VIS   VIVA   APPLIED   TO   MACHINES.        259 

the  fly-wheel  can,  in  the  last  case,  increase  the  power  of 
the  machine. 

But,  the  fly-wheel  being  always  a  heavy  piece,  caus- 
ing a  useless  consumption  of  work  by  its  friction  and  the 
resistance  of  the  air,  we  must  restrain  its  use  to  cases 
of  absolute  necessity,  and  give  it  a  suitable  limit  of 
weight. 


FKICTION. 

222. — We  usually  distinguish  two  kinds  of  friction? 
One,  called  friction  of  sliding,  is  produced  when  bodies 
slide  one  upon  the  other,  whence  it  results  that  the  primi- 
tive points  of  contact  are  found  ceaselessly  at  distances 
respectively  different  from  new  points  of  contact,  which 
is  expressed  in  saying  that  they  have  experienced  dis- 
placements, relatively  unequal,  and  in  opposite  directions. 
The  second  kind  of  friction,  improperly  called  rolling 
friction,  takes  place  when  bodies  roll  one  upon  the  other, 
when  the  distances  of  the  new  points  of  contact  from  the 
old  are  the  same  upon  both  bodies,  and  when  the  relative 
displacements  are  equal.  As  the  word  friction  implies, 
generally,  the  idea  of  sliding,  and  not  that  of  rolling,  it 
will  be  proper  to  admit  only  one  kind  of  friction,  that  of 
sliding,  and  to  designate  the  other  by  the  name  of  resist- 
ance to  rolling. 

223.  JKeview  of  ancient  experiments. — The  first  experi- 
ments known  upon  the  friction  of  sliding,  are  due  to 
Amontons,  and  are  inserted  in  the  Memoirs  of  the  An- 
cient Academy  of  Sciences,  1699.  This  philosopher  knew 
that  friction  was  independent  of  the  extent  of  surfaces, 
but  he  estimates  its  value  at  a  third  of  the  pressure  for 
wood,  iron,  brass,  lead,  etc.,  coated  with  lard,  which  is 
far  too  much. 


FRICTION.  261 

Coulomb,  officer  of  the  military  engineers,  and  some 
years  later  a  member  of  the  Institute,  presented,  in  1781, 
to  the  Academy  of  Sciences,  experiments  made  at  Boche- 
fort,  and  much  more  complete  than  those  of  Amontons'. 
The  apparatus  he  used  consisted  of  a  bench,  formed  of 
two  horizontal  timbers  6  feet  long,  upon  which  a  sledge 
loaded  with  weights  slid  by  the  action  of  a  weight  sus- 
pended to  a  cord,  which,  passing  over  a  fixed  pulley,  was 
attached  horizontally  to  the  sled. 

By  means  of  this  disposition,  Coulomb  at  once  de- 
termined the  effort  necessary  to  produce  motion  after  the 
bodies  had  remained  some  time  in  contact.  This  is  what 
he  called  the  resistance  or  friction  of  departure.  He  saw 
that  this  friction  was  proportional  to  the  pressure,  and  he 
expected  to  find  it  composed  of  one  part  proportioned  to 
the  extent  of  the  surface  of  contact,  which  he  termed  ad- 
hesion— and  of  another  part  independent  of  this  surface. 
He  then  sought  the  value  of  friction  during  motion,  and 
for  this  effect  he  observed,  with  a  stop-watch  of  half 
seconds,  the  time  employed  by  the  sled  in  running  suc- 
cessively the  first  three  feet  and  the  next  three  feet  of  its 
course. 

But  as  in  these  durations,  sometimes  equal  to  V  or  2", 
he  might  be  mistaken  by  a  half  second  at  the  end,  and 
also  at  the  commencement  of  the  experiment,  there  re- 
sulted very  great  uncertainties  which  did  not  admit  of 
establishing  his  conclusions  in  a  positive  manner,  and  we 
may  say  he  rather  conjectured  than  observed  the  laws 
which  he  inferred  from  his  experiments.  Still  he  admit- 
ted, that  generally,  friction  during  motion  is  : 

1st.  Proportional  to  the  pressure. 

2d.  That  it  is  independent  of  the  extent  of  the  surfaces 
of  contact. 

3d.  That  it  is  independent  of  the  velocity  of  motion, 
with  some  restrictions,  which  subsequent  experiments  did 
not  confirm. 


262 


FRICTION. 


Coulomb  also  first  established  the  fact,  that  for  com- 
pressible bodies,  the  friction  at  starting,  or  after  a  contact 
of  some  duration,  was  greater  than  it  was  after  the  first 
displacement. 

224.  Experiments  at  Metz. — The  uncertain  observa- 
tions, and  the  restrictions  adduced  by  Coulomb,  and  above 
all  the  more  general  use  of  inetals  in  the  construction  of 
machines,  called  for  a  new  series  of  experiments,  which  I 
made  at  Metz,  in  1831,  '32,  '33,  and  '34:,  by  means  of  new 
processes. 

225.  Summary  description  of  the  apparatus  used. — 
In  the  smelting  yards  of  this  ancient  foundry,  upon  a  flag- 
stone foundation,  and  at  the  side  of  a  trench,  (Fig.  98,) 


FIG.  93. 


was  established  a  horizontal  bed,  composed  of  two  parallel 
oak  beams  AA,  0.98ft-  square,  and  26.24"-  long,  connected 
and  supported  by  sleepers  3.28ft-  apart.  These  beams, 
which  jutted  about  4.26ft<  beyond  the  edge  of  the  trench, 
were  connected  with  four  uprights  BB,  between  which 
was  placed  a  platform  FF,  which  bore  the  pulley  for  pass- 
ing the  cord,  to  which  was  suspended  the  motive  weight, 


FRICTION.  263 

placed  in  a  box  K.  This  cord  was  fixed  horizontally  to  a 
sled  D,  charged  with  weights,  under  which  was  placed 
the  body  to  be  experimented  upon. 

The  cord,  instead  of  being  attached  directly  to  the 
sled,  was  fastened  to  the  front  plate  of  a  dynamometer 
with  a  style,  whose  flexure  measured  the  tension  of  the 
cord,  both  at  its  starting  and  during  its  motion. 

The  axis  of  the  pulley  had  a  copper  plate  H,  perfectly 
smooth  and  covered  with  a  sheet  of  paper.  Opposite  this 
plate,  clock-work  communicated  uniform  motion  to  a 
style,  formed  of  a  brush  filled  with  India  ink,  whose  point 
described  a  circle  0.459ft-  in  diameter.  The  parallelism 
of  the  plane  of  the  circle,  and  that  of  the  plate,  was  also 
perfectly  established  by  precise  methods,  and  the  contact 
of  the  brush  was  produced  or  interrupted  at  will. 

Upon  the  box  K  may  be  placed  two  others  for  holding 
weights,  which,  after  producing  the  motion,  may  at  a  cer- 
tain height  be  stopped  by  cleats,  so  that  the  motion  con- 
tinues only  in  virtue  of  the  load  and  weight  of  the  box  Q. 
By  this  means,  we  may  at  will  obtain,  with  the  box  Q 
alone,  an  accelerated  motion,  and  with  the  three  boxes,  a 
motion  at  first  accelerated,  then  uniform  or  retarded,  ac- 
cording as  the  weight  of  the  box  is  sufficient  to  overcome 
the  friction  or  is  inferior  to  this  resistance. 

Further  details  of  these  experiments  may  be  found  in 
the  "  Recueil  des  savants  etrangers,  tomes  IV  and  V,"  as 
well  as  in  a  memoir  published  in  1838,  by  M.  Carilian 
Gosury. 

226.  Examination  of  the  graphic  results  of  experi- 
ments.— We  may  conceive,  from  what  has  been  said  upon 
a  similar  apparatus  now  at  the  Conservatory  of  Arts  and 
Manufactures,  that  from  the  synchronism  of  the  two  mo- 
tions, the  one  of  the  style  being  uniform  and  with  a  known 
velocity,  and  the  other  unknown,  corresponding  in  a  con- 
stant ratio  with  the  spaces  described  by  the  sled,  there 


264:  FRICTION. 

must  result  a  curve  whose  abstract  will  give  us  the  law 
of  the  motion  of  the  sled.  "We  may  then,  by  this  ab- 
stract, form  a  table  of  spaces  described,  and  of  the  corre- 
sponding times,  and  construct  a  curve  whose  abscissae  are 
the  spaces  and  whose  ordinates  are  the  times.  The  curves 
thus  constructed  are  perfectly  continuous,  and  we  see,  as 
has  been  indicated  in  No.  81,  that  they  are  parabolas, 
that  is  to  say,  their  abscissae  are  proportional  to  the  square 
of  their  ordinates. 

From  the  fact  of  this  curve  being  a  parabola,  we  are 
justified  in  the  inference  that  the  motion  is  uniformly 
accelerated.  Now  the  motive  weight  being  constant,  the 
motive  force  producing  the  acceleration  of  motion  is  the 
excess  of  this  weight  above  the  friction,  and  since  this  ex- 
cess is  constant,  it  follows,  necessarily,  that  the  friction  is 
constant  and  independent  of  the  velocity. 

Experiments,  repeated  with  all  the  bodies  used  in 
the  construction  of  machines,  with  or  without  unguents, 
having  always  led  to  the  same  consequences,  we  are  au- 
thorized in  regarding  this  law  as  general,  at  least  within 
the  limits  of  the  velocity  of  observation ;  that  is  to  say, 
of  about  11.5ft-,  and  in  the  assumption  that  the  restrictions 
which  Coulomb  anticipated  have  no  existence  in  reality. 

227.  Formulae  employed  in  calculating  the  results  of 
experiments* — The  apparatus  which  we  have  just  described 
affords  a  simple  example  of  a  machine  in  which  the  mo- 
tion is  variable,  and  enables  us  to  apply  the  general  prin- 
ciples which  we  have  previously  pointed  out.  We  take 
advantage  of  it  to  show  the  method  of  procedure  in 
similar  cases. 

We  call  P  the  weight  of  the  descending  box,  including 
its  load  and  that  portion  of  the  cord  which  hangs  always 
under  the  pulley,  neglecting,  however,  the  quantity  by 
which  it  is  increased  in  its  descent,  which  seldom  exceeds 
2lb8- ;  T  the  tension  of  the  horizontal  strip ;  ^=13.T9lb9-  the 
weight  of  the  pulley. 


FRICTION.  265 

V,  the  angular  velocity  of  the  pulley  at  the  instant 
considered. 

Vi  the  quantity  by  which  the  velocity  varies  in  an  ele- 
ment of  time  t. 

I=.  04551  the  moment  of  inertia  of  the  pulley  and  of 
the  pieces  turning  with  it. 

f  =0.164:  a  ratio  determined  by  special  experiments, 
of  the  friction  to  the  pressure,  for  the  iron  axle  of  the 
pulley  and  the  ash-  wood  cushions  greased  ;  B=0.032T 
the  rigidity  of  the  twisted  cord,  determined  also  by 
especial  experiments. 

N  the  pressure  of  the  axle  of  the  pulley  upon  the 
journals. 

r  the  radius  of  the  pulley. 

r'  the  radius  of  its  journals. 

If  we  refer  to  the  principles  laid  down  in  No.  186, 
upon  the  motion  of  variable  rotation,  we  shall  see  that  at 
each  instant  of  the  motion  of  the  pulley,  the  sum  of  the 
moments  of  the  exterior  forces  must  be  equal  to  the  sum 
of  the  moments  of  the  forces  of  inertia. 

Now,  the  sum  of  the  moments  of  the  exterior  forces  is 

Pr—  Tr—Rr—N  .  rf. 


The  sum  of  the  moments  of  the  forces  of  inertia  answering 
to  a  velocity  vl  of  angular  velocity  is  easily  found  ;  for, 
one  of  these  forces,  relative  to  a  molecule  of  the  mass  m, 

qj  M 

situated  at  a  distance  r^  being  m  .  -i-^,  its  moment  in  re- 

t 
tf 
spect  to  the  axis  is  mr?—  l,  and  the  sum  of  the  similar  mo- 

u 

ments  is  I-1,  for  all  parts  turning  around  the  axis. 
t 

The  moment  of  inertia  of  the  weight  P  is  —  —  r,  and 

9    * 
must  be  added  to  the  preceding  ;  we  have  then,  at  each 

instant  of  variable  motion  of  the  pulley,  the  relation 


266  FRICTION. 

The  pressure  "N  upon  the  axle  of  the  pulley  being  the  re- 
sultant of  two  perpendicular  forces,  the  one  horizontal 
equal  to  the  tension  T,  the  other  vertical  and  equal  to  the 
weight  P  of  the  box,  increased  by  the  weight  of  the  pul- 
ley, and  diminished  by  the  force  of  inertia  -  — ,  which  is 

9   t 

developed  in  the  acceleration  of  the  vertical  motion  of  the 
weight  P,  and  is  opposed  to  its  acceleration ;  we  have 
then 


Now,  according  to  an  algebraic  theorem  of  Poncelet,  the 
value  of  a  radical  of  the  form  yV+52,  in  which  we 
know  beforehand  that  a  >  5  is  given  to  nearly  -^j  by  the 
formula  0.96  a+QAl.  In  applying  it  to  the  case  in  hand, 

where  we  have  always  P+#  ---  ->T,  since  the  weight 

^  * 

P  exceeds  the  resistance  T  and  the  friction  of  the  sled, 
we  have  to  -£j  nearly 


$T=0.96  JP  +2—  -^ 
'  9    * 


The  relation  of  the  equality  of  moments  becomes  then,  in 
making  K=0.032T, 

.032Tr—  0.96//  {  P+#—  -  ^  (  —  0.4//T 
'  t    } 


P 


and  in  deriving  from  this  equation  of  the  first  degree  the 


FEICTION.  267 

value  of  T,  the  tension  of  the  horizontal  strip  of  the  cord, 
we  find 

T  j  1+0.032+0.4?^-'  1  =P  1 1—0.96^  i  —  0.96/^ 


g    t   \  r   )       r*  t 

In  substituting  for  the  known  quantities  their  values, 
which  are 

/=0.164,  r'= 0.030512"-,  r=0.36417ft-,  I=.04551, 

whence  ^=0.34317, 

if 

we  have  for  the  practical  formula  which  gives  the  tension 
T,  when  we  know  the  weight  P  of  the  box, 

T=0.95  \  P— (.34685+-)^  [  — 0.1753lb3- 
(         v  git    > 

When  experiment  has  demonstrated  that  the  accelera- 

n\  M 

tion  -i-  is  constant,  and  the  abstract  of  the  curves,  in  giv- 
t 

ing  their  equation  T3=2CE,  shall  have  furnished  for  the 

-« 

acceleration  the  value  —=^1  in  calling  20  the  parameter 
t       O 

of  the  parabola,  we  shall  have  all  the  elements  required 
to  calculate  the  value  of  the  tension  of  the  cord  in  the  ex- 
periment. It  will  be 

T=0.95 1  P— (.34685+?U  I  —  .1753lbs- 
(        V  g'Vy 

Qy-tf* 

When  the  motion  is  uniform  the  acceleration  —  =7, 

t      *-> 

is  zero,  and  the  above  formula  is  reduced  to 
T=0.95P— 0.1753lb% 


268  FRICTION. 

or  simply  T=0.95P,  on  account  of  the  small  value  of  the 
second  term  .lT53lbs- 

In  extracting  directly  from  the  curves  of  tension  of 
the  dynamometer,  the  values  of  T  relative  to  more  than 
forty  experiments,  in  which  the  loads  have  varied  from 
26  to  209  \  pounds,  we  have  found  that  the  ratio  of  the  ten- 
sion to  the  load,  thus  furnishing  a  direct  measurement, 
was  at  0.96,  which  shows  that  all  the  data  introduced  in 
the  above  formula  leads  to  a  result  which  accords  with 
this  measure,  within  very  satisfactory  limits  of  correctness. 

228.  Relations  between  the  tension  of  the  cord  and  the 
friction  of  the  sled.  —  Knowing  the  tension  of  the  cord  T, 
by  means  of  the  dynamometer,  or  having  calculated  it 
by  the  preceding  formula,  it  is  quite  easy  to  deduce  the 
value  sought,  of  the  friction  of  the  sled,  in  applying 
directly  the  principle  of  action  equal  and  opposite  to  re- 
action. In  fact,  the  tension  T,  and  the  friction  sought  F, 
are  two  external  forces  with  opposite  directions,  whose 
diiference  T  —  F  produces  the  motion  of  acceleration  of 
the  sled.  On  the  other  hand,  the  resistance  which  the 
inertia  of  the  weight  Q  of  the  sled  opposes  to  this  accel- 

eration is  (No.  62)  %£. 


We  have  then  for  the  equality  of  action  and  reaction, 


whence 

P_T     Q    1 


When,  by  direct  observation,  or  by  the  formula  of  the 
preceding  number,  we  shall  have  determined  the  tension 
of  the  cord,  we  must  for  the  value  of  the  friction  subtract 

from  it  the  quantity  —  -  ,  easily  calculated  when  we  know 


FRICTION.  269 

by  the  abstract  the  parameter  2C  of  the  curve  of  mo- 
tion. 

Such  is  the  method  which  was  adopted  for  the  calcu- 
lation of  all  the  experiments  where  the  motion  was  accel- 
erated ;  as  for  those  where  the  motion  is  uniform,  we 
have  simply  F=T. 

We  see  that  the  law  of  the  motion  being  once  known 
by  the  abstract  of  the  curves,  and  being  that  of  a  uni- 
formly accelerated  motion,  we  may,  after  having  proven 
the  constancy  and  the  generality  of  this  law,  pass  to  the 
use  of  the  dynamometer,  and  rest  content  with  the  indi- 
cations of  the  chronometric  apparatus. 

229.  Results  of  experiments. — I  give  here,  as  exam- 
ples of  the  results  obtained,  some  of  the  tables  inserted  in 
my  memoirs,  successively  presented  to  the  Institute  and 
inserted  in  the  "  Kecueil  des  savants  etrangers,"  and  as  an 
example  of  the  application  of  the  preceding  formula,  I 
select  the  second  experiment  of  the  first  of  these  tables, 
relative  to  the  friction  of  oak,  sliding  upon  oak  without 
unguent,  with  the  fibres  parallel  to  the  direction  of  the 
motion. 

In  this  experiment  we  have 

Q=295.22lbs-    P=203.38lbs- 
The  trace  of  the  curve  gives  for  the  parameter 

2C=0.6339f% 
whence 


and  consequently  the  tension 

T=0.95  j  P— (0.34685+?)  ^  1—  .1753=173.05lb8- 


270  FRICTION. 

The  other  formula  gives  for  the  value  of  friction 


The  ratio  of  the  friction  to  the  pressure  is  here  then 


_. 
Q-295T2- 

TABLE. 

Experiments  upon  the  friction  of  oak  upon  oak  without 
unguents  —  the  fibres  of  the  wood  being  parallel  to  the 
direction  of  motion. 


g 

" 

g 

Velocity  of  motion. 

1 

'i 

1 

. 

1 

g 

H 

1 

•0 

ll 

1 
•8 

1 

IH.O 

g 

1 

fe 

1  § 

i 

I 

l!-l 

1 

fi 

£ 

••3 

"1 

z 

*3 

&3 

1 

If  8 

o 
1 

b 

1 

F 

1*3 

Q 

P 

T 

F 

Q 

Sq.  ft. 

lb» 

Ibs. 

Ibs. 

ft. 

Ibs. 

ft. 

ft. 

295.22 

148.58 

141.15 



141.15 

0.477 

2.264 

295.22 

203.38 

173.02 

0.634 

3.123 

144.1 

0.488 

'  7.77 

333.52 

171.03 

162.48 

162.48 

0.487 



0  TOO 

970.63 

504.32 

479.10 

479.44 

0.491 

1.345 



6,  iVO  • 

970.63 

610.01 

536.64 

0.850 

2.352 

466.41 

0.480 

6.726 

1499.13 

930.23 

819.18 

0.862 

2.820 

709.33 

0.472 

6.693 

2291.56 

1273.69 

1164.91 

1.688 

1.184 

1080.60 

0.471 

6.299 

2291.56 

1114  91 

1059  16 

1059.16 

0462 

8.511 

102.09 

64.95 

5417 

1.914 

1.044 

50.86 

0.49S 

4.495 

108.53 

56.59 

53.77 

53.77 

0.496 

4.20' 



120.55 

6290 

5975 

59.75 

0.495 

4.92 

120.55 

98.39 

76.44 

0.384 

5.208 

56.95 

0.472 

10.072 

1.062  • 

226.81 

186.83 

152.57 

0.472 

4.237 

110.38 

0.486 

8.924 

227.63 

182.64 

117.77 

1.054 

1.897 

104.36 

0.458 

6.102 

332.76 

162.72 

154.58 



154.58 

0.464 

4.ioi 



440.03 

211.37 

200.80 

200.84 

0.456 

2.001 



440.24 

210.45 

199.93 



199.93 

0.454 

2.789 



(    21567 

10862 

103.19 

103.19 

0.478 

3.478 

0.33   4    321.47 

175.49 

164.74 

0.933 

2.145 

133.34 

0.414 

6.918 

(    604.06 

468.80 

389.44 

0.506 

8.952 

293.51 

0.484 

8.858 

When  the  motion  is  uniform,  as  in  the  sixteenth  ex- 
periment of  the  same  table,  we  have  simply,  for 

P=211.37lbs-, 


FKICTION. 


2T1 


An  examination  of  the  different  tables  relating  to 
these  very  variable  cases,  completely  establishes  the  laws 
of  friction,  which  are  to  be  used  in  the  motion  between 
the  greater  part  of  materials  employed  in  the  arts.  The 
results  of  all  the  other  experiments  agree  with  those 
which  we  limit  ourselves  to  reporting  here. 

TABLE. 

Experiments  upon  the  friction  of  Elm  upon  Oak,  without 
unguents — the  fibres  of  the  wood  ~being  parallel  to  direc- 
tion of  motion. 


| 

i 

2 

•s 

* 

b 

L 

ij 

1 
f  g 

J-,0 

S 

a 

**-  s 

0     * 

1 

£ 

.1 

ii 

I 

1 

1 

Ii 

I' 

m 

i 

S* 

9 

£ 

Sq.  ft. 

Ibs. 

Ibs. 

Ibs. 

ft 

Ibs. 

ft. 

260.05 

161.31 

139.19 

0.732 

2.734 

117.18 

0.45 

7.55 

260.05 

187.42 

153.06 

0.469 

4.261 

118.88 

0.45 

9.45 

921.38 

506.69 

450.40 

0.984 

2.031 

892.27 

0.43 

8.60 

921.88 

480.24 

440.46 

1.859 

1.075 

408.78 

0.44 

4.62 

1.838- 

921.38 
921.38 

454.13 
664.42 

416.77 
525.72 

1.902 
0.377 

1.051 
5.291 

386.62 
374.53 

0.42 
0.41 

4.48 
12.46 

1980.10 

1113.83 

976.94 

0.802 

2.494 

821.76 

0.42 

7.41 

1980.10 

1007.77 

927.09 

1.993 

1.003 

865.54 

0.44 

4.04 

1980.10 

1118.83 

911.42 

1.206 

1.657 

787.42 

0.40 

5.68 

1980.10 

1298.70 

1104.86 

0.600 

8.330 

899.99 

0.45 

8.10 

244.81 

135.42 

122.86 

1.414 

1.414 

108.93 

0.45 

5.25 

.063- 

389.58 

811.19 

240.06 

0.347 

5.756 

171.43 

0.44 

10.50 

917.79 

479.76 

489.82 

1.734 

1.153 

408.60 

0.44 

4.76 

Mean 0.434 


272 


FRICTION. 


TABLE. 

Experiments  upon  the  friction  of  soft  oolitic  Limestone  of 
Jaumont,  near  Metz,  upon  stone  of  the  same  kind  with- 
out unguent. 


, 

| 

j 

o 

•s 

•s 

L 

I 

If 

is! 

i,<; 

J, 

j 

Si 

1 

£ 

J§ 

1! 

i 

1 

£ 

|  1 

I5 

1 

1 

EH  ^ 

••- 

£ 

Sq.ft. 

Ibs. 

)ba. 

Ibs. 

ft. 

Ibs. 

ft. 

314.04 

254.18 

222.40 

0.829 

2.412 

198.89 

0.633 

6.890 

814.04 

254.18 

218.36 

0.682 

2.929 

187.60 

0.597 

7.579 

0.861  - 

1264.18 

999.63 

853.54 

0.621 

8.216 

727.50 

0.575 

7.940 

1274.94 

1034.92 

859.41 

0.499 

4.001 

700.86 

0.549 

8.858 

1274.94 

1034.92 

859.41 

0.499 

4.001 

700.86 

0.549 

8.858 

Mean. 


.0.580 


0.499 


309.56 
331.62 
1257.50 
1257.50 
1257.50 

293.88 
293.88 
1034.92 
1140.78 
1140.78 

245.40 
244.65 
925.13 
943.99 
924.32 

0.536 
0.524 
1.066 
0.488 
0.426 

8.725 
3.815 
1.874 
4.101 
4.687 

209.56 
207.97 
851.95 
783.89 
741.28 

0.677 
0,627 
0.677 
0.623 
0.589 

8.498 
8.662 
6.070 
8.990 
9.613 

.0.639 


298.40 

240.95 

218.30 

1.426 

1.402 

205.31 

0.688 

5.249 

298.40 

240.95 

211.02 

0.841 

2.877 

189.01 

0.633 

6.824 

298.40 

293.88 

239.18 

0.451 

4.433 

198.10 

0.664 

9.350 

597.93 

465.91 

421.45 

1.841 

1.491 

393.79 

0.659 

5.413 

597.93 

465.91 

431.15 

2.499 

0.800 

416.28 

0.696 

3.970 

597.93 

571.77 

485.40 

0.597 

3.347 

423.25 

0.709 

8.104 

.0.675 


General  mean. 


When  the  soft  limestone  slides  upon  soft  limestone, 
and  especially  when  the  moving  body  rests  upon  surfaces 
of  small  area,  the  latter  are  destroyed  rapidly  during  the 
experiment.  This  circumstance,  and  the  presence  of  the 
dust  powder  resulting  from  it,  have  not  changed  the  laws 
observed. 


FRICTION. 


273 


TABLE. 

Experiments  upon  the  friction  of  strong  leather ',  tanned, 
and  placed  flatwise  upon  cast-iron. 


a 

1 

1 

i 

2 

JS 

•s 

fi 

1 

2 

y 

J 

1 

L 

| 

li 

li 

08 

i 

J 

y 

1 

I 

5 

E 

o2 

I" 

I 

i 

P 

§ 

1 

1 

Sq.  ft. 

0.4155 

a 

Iba. 
471.02 
1106.42 

Ib*. 
320.35 
637.94 

Ibs. 
291.83 
606.04 

ft. 
1.548 

1.292 

272.75 
606.04 

0.579 

0.547 

5.02 

Mean 0.563 


0.4155 


291.01 
291.01 
I     291.01 
I  1115.03 


188.02 
161.55 
135.08 
977.58 


154.78 
183.85 
118.83 
689.54 


0.497 


0.926 
0.244 


4.024 
3.816 
2.159 
8.196 


118.63 
96.44 
99.49 

407.11 


Mean. 


0.408 
0.842 
0.842 
0.865 

.0.364 


8.66 

6.56 

12.70 


0.4155 


£  r  1114.10 

o  J  1114.10 

3  1  1114.10 

[  1114.10 


0.4155  §  j 


298.49 

299.17 

1114.10 

1114.10 


214.48 
820.36 
426.10 


92.19 
148.32 
214.48 


193.80 
198.54 
279.52 
350.41 


87.30 

76.29 

140.91 

196.22 


2.042 
2.584 
0.795 
0.475 


0,548 
1.804 


0.979 
0.776 
2.516 
4.210 


168.21 
172.38 
192.43 


0.146 
0.155 
0.172 
0.164 


Mean 0.159 


8.649 


1.108 


87.29 

42.52 

140.91 

157.93 


Mean. 


0.124 
0.142 
0.126 
0.141 

.0.133 


4.58 

8.87 
7.09 


8.46 
4.59 


tin 


1114.10  I    320.35 
478.92  I    185.08 


294.48 
123.77 


2.011 
1.950 


0.944  I  260.07 
1.025  I  108.66 


0.227 
Mean...     . .  .2.30 


4.66 
4.66 


Though  leather  is  a  soft  and  very  compressible  sub- 
stance, its  friction  is  none  the  less  proportional  to  the 
pressure,  and  independent  of  the  velocity,  throughout  the 
whole  range  of  the  experiments. 

18 


274: 


FRICTION. 


TABLE. 

Experiments  upon  the  friction  of  brass  upon  oak,  without 
unguent. — Fibres  of  wood  parallel  to  the  direction  of 
motion. 


• 

fco 

1 

a 

i 

•s 

1 

1 

.11 

I  S 

8 
| 

1 

«  "*  |O 

J~" 

| 
•1 

'II 

| 

1 

£ 

I 

1 

(2 

fi 

1 

1 

i 

H 

> 

Sq.  ft. 

Ibs. 

Ibs. 

Ibs. 

ft. 

Ibs. 

ft. 

257.13 

161.46 

153.36 

.... 

153.39 

0.60 

.... 

257.13 

161.61 

153.61 

153.54 

0.60 

.488    - 

1539.90 
1539.90 

981.99 
1114.32 

932.69 

1068.80 

1.548 

1.291 

932.89 
1007.79 

0.60 
0.65 

1539.90 

1273.11 

1101.97 

0.707 

2.828 

967.05 

0.62 

748 

1989.83 

1378.97 

1290.72 

4.846 

0.460 

1262.61 

0.63 

305 

248.81 

161.72 

153.60 

153.61 

0.61 

248.49 

188.86 

169.56 

1.283 

i.558 

157.58 

0.63 

5.2l' 

0.141     - 

763.97 

532.07 

487.11 

1.956 

1.022 

462.99 

0.60 

4.92 

1531.26 

981.89 

932.69 

932.89 

0.61 

1531.26 

1273.11 

1103.69 

6.719 

2.780 

971.73 

0.63 

7.51 

Mean. 


.0.616 


For  the  experiments  where  we  have  not  indicated  the 
value  of  the  parameter  of  the  law  of  motion,  and  that  of 
the  acceleration,  the  motion  was  slow  and  somewhat  un- 
certain. 

The  results  contained  in  this  table  confirm  the  three 
laws  before  enumerated,  but  we  remark  that  the  mean 
value  of  the  friction,  which  is  here  616,  is  more  consid- 
erable than  in  the  case  of  oak  rubbing  against  oak,  or 
than  that  of  elm  upon  oak,  for  which  the  results  are  con- 
signed to  the  tables  of  pages  270  and  271. 

"We  shall  see,  by  the  following  table,  that  the  coeffi- 
cient diminishes  considerably  when  the  friction  occurs 
between  two  metallic  surfaces. 


FRICTION.  275 

Experiments  upon  the  friction  of  cast  iron  upon  cast  iron. 


1 

1 

£ 

A 

I 

1 

i 

I 

1 

J 

I* 

*s 

1 

I 

£ 

J 

| 

| 

J 

g- 

s 

« 

%-. 

5     • 

J 

*d 

1  o 

^  !r" 

t£» 

§ 

s 

1 

P 

1 

>a 

1 
1 

of  the  cord 

£ 

! 

£ 

tio  of  fricti 

locity  at  9. 

1 

1 

* 

£ 

H 

Sq.  ft. 

Ibs. 
496.10 

Ibg. 

108.62 

lb«. 
95.78 

0.993 

2.012 

11,8. 

64.49 

0.130 

ft. 
6.37 

vj. 

496.10 

135.09 

113.88 

0.585 

3.417 

60.79 

0.122 

8.20 

OO 

a 

1091.14 

820.37 

283.82 

0.938 

2.130 

211.15 

0.193 

6.50 

0.3874 

1  - 

1091.14 
1104.80 

426.08 
174.79 

336.38 
166.05 

0.378 

5.291 

157.18 
166.05 

0.144 
0.150 

10.17 

Slow. 

ft 

4412.70 

796.73 

745.58 

4.267 

0.468 

681.74 

0.154 

8.25 

4412.70 

929.06 

865.85 

8.816 

0.604 

783.54 

0.177 

8.48 

4412.70 

1054.77 

949.52 

1.158 

1.726 

712.94 

0.161 

5.81 

Mean....  0.154 


0.3874 


0.3874 


0.3874  .2 


0.8874 


^    f  110437 

899.74 

861.17 

1.402 

1.426 

312.82 

0.282 

8.90 

«  J  110437 

505.61 

432.96 

0.646 

3.095 

824.60 

0.298 

9.25 

£  1  2202,70 

770.26 

731.36 

731.36 

0.832 

Uniform. 

?    [2202.70 

876.13 

806.43 

2.036 

0.982 

739.30 

0.386 

458 

Mean....  0.811 

x    (  1091.14 

201.25 

191.15 

191.15 

0.175 

Slow. 

J  •<  1091.14 

820.87 

287.77 

1.251 

1.598 

281.00 

0.211 

5.68 

(  1091.14 

373.30 

821.78 

0.'695 

2.878 

224.47 

0.205 

7.09 

Mean....  0.197 

f   496.10 

52.49 

49.87 

49.87 

0.100 

Slow. 

496.10 

78.96 

65.48 

1.950 

1.024 

50.40 

0.101 

456 

1103.43 

108.64 

103.17 

103.17 

0.093 

Slow. 

^ 

1103.43 

201.25 

179.20 

l".060 

1.885 

114.64 

0.104 

6.17 

o 

1103.43 

240.95 

212.87 

0.939 

2.130 

117.81 

0.106 

6.47 

"3 

2214.98 

293.88 

271.14 

2.286 

0.875 

211.30 

0.095 

420 

£-1 

2214.98 

293.88 

274.54 

4.023 

0.497 

243.84 

0.109 

8.08 

2214.98 

426.10 

379.70 

0.999 

2.000 

243.33 

0.109 

6.30 

6185.82 

62470 

593.47 

593.47 

0.096 

.... 

Very  slow. 

1108.14 

108.62 

103.17 

103.17 

0.093 

Slow. 

Mean....  0.101 

129.48 

118.70 

2.011 

0.994 

84.62 

0.076 

4.53 

129.48 

118.13 

1.767 

1.131 

79.44 

0.071 

4.72 

133.89 

12119 

1.395 

1.432 

72.16 

0.065 

5.61 

133.89 

120.99 

1.414 

1.414 

72.54 

0.066 

5.58 

frj 

138.31 

126.29 

1.767 

1131 

85.82 

0.077 

4.59 

s 

1103.43 

138.31 

124.41 

1.295 

1.544 

71.55 

0.065 

5.51 

^ 

138.35 

123.55 

1.341 

1.491 

72.71 

0.066 

5.44 

138.35 

124.41 

1.295 

1.544 

71.55 

0.065 

5.51 

193.44 

168.15 

0.783 

2.553 

80.65 

0.073 

7.12 

193.44 

167.07 

0.731 

2.734 

72.94 

0.066 

7.28 

, 

193.44 

168.92 

0.823 

2.430 

85.6S 

0.078 

6.82 

Mean....  0.070 


276  FEICTION. 

This  table,  besides  verifying  tlie  laws  of  the  propor- 
tionality of  the  friction  to  the  pressure,  and  its  independ- 
ence of  the  velocity,  shows  that  water  rather  increases 
than  diminishes  the  friction  of  cast-iron.  "We  see  also 
that  tallow,  somewhat  hard,  does  not  reduce  the  friction  so 
much  as  lard. 

230.  Consequences  of  the  experiments. — The  experi- 
ments made  by  me  upon  the   friction  proper  of  plane 
surfaces  upon  each  other,  comprise  179  series,  answering 
to  different  cases,  according  to  the  nature  or  condition  of 
the  surfaces  in  contact ;  and  they  all,  without  exception, 
lead  to  the  following  results : 

The  friction  during  the  motion  is : 
1st.  Proportional  to  the  pressure. 
2d.  Independent  of  the  area  of  the  surfaces  of  con- 
tact. 

3d.  Independent  of  the  velocity  of  motion. 

231.  Experiments  upon  the  friction  at  starting,  or 
when  the  surfaces  have  been  some  time  in  contact. — The 
same   apparatus  has  served  for  the   experiments  upon 
friction  at  the  start,  or  after  a  prolonged  contact,  whose 
aim  was  to  establish  in  wrhat  cases  there  is  a  notable  dif- 
ference  between  it  and  that  produced  during  motion. 
This  difference,  which,  according  to  the  case,  arises  from 
very  different  causes,  may  in  general  be  attributed  to  the 
reciprocal  compression  of  the  bodies  upon  each  other, 
and  to  a  kind  of  gearing  of  their  elements.     The  time  or 
duration  of  the  compression  probably  exerts  an  influence 
upon  the  intensity  of  the  resistance  opposed  by  their  sur- 
faces to  sliding.     But,  generally,  this  resistance  reaches 
its  maximum  at  the  end  of  a  very  short  period. 

232.  Results  of  experiments. — We  publish  here  the  re- 
sults of  some  experiments  which  we  have  made. 


FRICTION. 


277 


TABLE. 

Experiments  upon  the  friction  of  oak  upon  oak,  without 
unguents,  when  the  surfaces  have  ~been  some  time  in  con- 
tact. The  fibres  of  the  sliding  pieces  leing  perpendicu- 
lar to  those  of  the  sleeper. 


Extent  of  the  surface 
of  contact. 

Pressure 
Q. 

Motive  effort  or 
friction 
P. 

Ratio  of  the  friction 
to  the  pressure/ 

Sq.ft. 

Ibs. 

Ibs. 

120.55 

67.15 

0.55 

282.49 

150.23 

0.53 

0.947 

495.01 

252.34 

0.51 

1995.23 

1171.10 

0.58 

2526.65 

1287.16 

0.51 

389.35 

203.80 

0.52 

0.043 

402.98 

212.44 

0.53 

1461.08 

854.77 

0.58 

Mean. 


.0.54 


The  friction  seems  to  be  proportional  to  the  pressure, 
which  varied  from  120lbs-  to  2526lb%  and  independent  of 
the  surfaces  of  contact,  which  varied  in  the  ratio  of  1  to 
22,  the  smallest  being  .043sq-  f%  and  the  greatest  0-947S<1- ft' ; 
this  last  value  exceeds  those  usually  employed  for  sliding 
surfaces  in  mechanical  constructions. 

The  ratio  of  the  friction  to  the  pressure  is  here  raised 
to  0.54,  while  it  was  only  0.48  during  the  motion,  as  was 
the  result  of  the  table  page  270.  The  friction  at  the  start 
is  raised  then  about  an  eighth  above  that  which  we  first 
considered.  A  similar  increase  occurs  in  all  similar 
cases. 


278 


FEIOTION. 


TABLE. 

Experiments  upon  the  friction  of  oak  upon  oak,  without 
unguents,  when  the  surfaces  have  been  some  time  in  con- 
tact. The  sliding  pieces  have  their  fibres  vertical,  those 
of  the  fixed  pieces  are  horizontal  and  parallel  to  the 
direction  of  motion. 


Extent  of  the   sur- 
face of  contact 

Pressure 
Q. 

Motive  effort  or 
friction  F. 

Ratio  of  friction 
to  pressure/ 

Time 
of  contact. 

Sq.  ft. 

Ibs. 
432.12 

Ibs. 
184.88 

0.427 

5  to  6" 

432.12 

184.88 

0.427 

10' 

432.12T 

157.43 

0.364 

1' 

696.77 

354.59 

0.509 

6' 

696.77 

304.31 

0.436 

30" 

6845 

696.77 

342.03 

0.498 

8  to  10' 

882.01 

405.32 

0.459 

8  to  10' 

1106.99 

555.73 

0.502 

10' 

1106.99 

430.03 

0.388 

5  to  6" 

2205.30 

810.24 

0.367 

15' 

2205.30 

882.60 

0.400 

10' 

Mean 0.434 


This  table  shows  that  for  wood,  the  friction  at  the 
start  presents  for  equal  surfaces  and  pressures  great  dif- 
ferences from  one  experiment  to  another,  and  that  the 
resistance  attains  its  maximum  in  a  short  time  of  contact, 
which  seems  not  to  exceed  some  seconds.  We  in  fact  see 
that  the  figures  answering  to  5  and  6  seconds  are  not 
inferior  to  those  relating  to  a  contact  of  15  minutes,  the 
longest  of  any  recorded  in  the  table. 

The  mean  value  of  the  ratio  f  of  friction  to  the  pres- 
sure is  0.434,  but  it  would  be  well  in  application  to  reckon 
it  at  0.48  or  even  0.50. 


FRICTION. 


279 


TABLE. 

Experiments  upon  the  friction  of  oolitic  limestone  upon 
oolitic  limestone,  when  the  surfaces  have  leenfor  some 
time  in  contact. 


Surface  of  contact         Pressure 

Motive  effort 
or  friction  F. 

Eatio  of  friction 
to  the  pressure 

Time  of  contact. 

Sq.ft. 
0.8611 

Ibs. 
{       314.01 
330.85 
-       1162.72 
1274.93 
1274.93 

Ibs. 
228.88 
239.25 
949.64 
932.87 
958.02 

0.728 
0.723 
0.752 
0.731 
0.751 

15' 
15' 
15' 
5  to  6" 
5  to  6" 

Mean  

0.737 

0.4992 

l        309.55 
\      1257.49 
(      1257.49 

228.88 
983.16 
983.16 

0.739 
0.781 
0.781 

2' 
10' 
1' 

Mean  

0.783 

Edges 
rounded 

1       298.38 
J       602.32 

228.88 
442.58 

0.774 
0.740 

2' 
5  to  6" 

Mean., 
General  Mean.. 


.0.757 
.0.740 


We  still  see  by  these  experiments  that  the  friction  at 
starting,  as  well  as  the  friction  in  motion,  is  independent 
of  the  extent  of  the  surface  of  contact,  and  is  proportional 
to  the  pressures.  This  conclusion,  and  even  the  value 
deduced  from  the  above  experiments,  has  been  since  con- 
firmed by  results  obtained  in  similar  cases  by  M.  Boistard, 
Engineer  of  Roads  and  Bridges,  in  1822. 

These  figures  moreover  differ  so  little  from  each  other, 
that  we  may  place  all  confidence  in  the  general  mean 
0.74,  and  employ  it  in  all  similar  cases. 


280 


FRICTION. 


TABLE. 

Experiments  upon  the  friction  of  oolitic  limestone  upon 
oolitic  limestone,  when  the  surfaces  have  been  some  time 
in  contact,  with  a  bed  of  fresh  mortar. 


Surface  of  contact. 

Pressure 
Q. 

Motive  effort  or 
friction  F. 

Eatio  of  friction 
to  pressure/ 

Time  of  contact. 

Sq.ft. 

Ibs. 

Ibs. 

325.66 

253.98 

0.780 

10' 

506.08 

404.87 

0.800 

10' 

783.98 
783.98 

580.87 
608.22 

0.740 
0.773 

15' 
10' 

783.98 

555.73 

0.709 

10' 

1167.73 

983.16 

0.841 

15' 

Mean  

0.773 

'       309.55 

239.21 

0.772 

10' 

489.97 

379.74 

0.775 

10' 

781.10 

568.30 

0.727 

10' 

0.4992 

1164.86 

807.15 

0.792 

15' 

1164.86 

907.74 

0.779 

10' 

1169.27 

807.15 

0.690 

10' 

1548.61 

1159.17 

0.748 

15' 

Mean  

0.745 

f       319.82 

254.02 

0-794 

10' 

niAo«                   500.25 
0.1636          1        79137 

304.30 
480.26 

0.608 
0.607 

10' 
10' 

1161.90 

731.69 

0.629 

15' 

Mean. 
General  mean. 


.0.659 
.0.735 


These  experiments  show  that  the  friction  at  starting  is 
for  these  stones,  very  nearly  the  same,  with  the  interposi- 
tion of  mortar  as  without. 

In  recapitulating,  recent  trials  have  caused  us  to  see 
that  the  friction  at  the  moment  of  starting,  and  after  a 
very  short  time  of  contact,  is : 

1st.  Proportional  to  the  pressure. 

2d.  Independent  of  the  area  of  the  surfaces  of  con- 


FRICTION.  281 

tact ;  and  that  furthermore,  for  compressible  bodies,  it  is 
notably  much  greater  than  that  which  takes  place  during 
motion. 

233.  Observation  relative  to  the  expulsion  of  unguents 
under  heavy  pressures,  and  by  prolonged  contact. — "We 
have  observed  metallic  bodies  with  unguents  of  grease  or 
oil,  under  very  great  pressure,  compared  to  their  surfaces, 
and  find,  after  a  contact  of  some  duration,  that  the  un- 
guents are  expelled,  so  that  the  surfaces  are  simply  in  an 
unctuous  state,  and  thus  have  double  the  friction  of  sur- 
faces well  greased.    This  shows  us  why  the  effort  required 
to  put  certain  machines  in  motion  is,  disregarding  the  in- 
fluence of  inertia,  often  much  greater  than  that  required 
for  maintaining  a  rapid  motion,  and  proves  that,  for  an 
experimental  appreciation  of  the  friction  of  machines  in 
motion,  we  need  not,  as  is  sometimes  done,  make  use  of 
the  same  methods  as  for  machines  starting  from  repose. 

234.  Influence  of  vibrations  upon  the  friction  at  start- 
ing.— Another   remarkable    circumstance  noted  in  the 
experiments  at  Metz  is,  that  when  a  compressible  body  is 
solicited  to  slide  by  an  effort  capable  of  overcoming  the 
friction  of  motion,  but  inferior  to  the  friction  at  starting, 
a  simple  vibration,  produced  by  an  external  and  appa- 
rently a  slight  cause,  may  determine  the  motion.    Thus, 
for  oak  rubbing  on  oak,  the  friction  at  starting  is  0.680  of 
the  pressure,  and  the  friction  during  motion  is  0.480 ;  so 
that,  to  produce  the  motion  of  a  weight  of  2205lbs-  it  is 
necessary  then  to  exert  an  effort  of  1500lb%  while  there  is 
only  needed  1059lbs-  to  maintain  it.     Still,  under  an  effort 
equal,  or  a  little  above  1059lb%  and  by  the  effort  of  a 
vibration,  the  body  may  be  started. 

This  important  observation  applies  to  constructions 
always  more  or  less  exposed  to  vibrations,  and  shows  that, 
if  in  the  calculations  for  machines  for  producing  motion, 
we  should  take  into  account  the  greatest  value  of  the 


282  FRICTION. 

friction,  we  should  in  those  relating  to  the  stability  of 
constructions,  on  the  other  hand,  introduce  its  smallest 
value,  that  for  motion.  It  seems,  finally,  to  explain  how 
it  sometimes  happens,  that  buildings,  for  the  stability  of 
which  no  uneasiness  was  felt,  have  fallen  at  the  passing- 
of  a  wagon,  and  how  the  firing  of  salutes  from  a  breach 
battery  may,  at  certain  times,  accelerate  the  fall  of  a  ram- 
part or  a  building. 

235.  Influence  of  unguents.— Tat  unguents  considera- 
bly diminish  friction,  and  the  consequent  wear  of  surfaces. 
But  from  the  observations  made  (No.  230),  we  see  that 
though  the  friction  is  in  itself  independent  of  the  extent 
of  the  surfaces,  it  is  well  to  proportion  them  to  the  pres- 
sures they  are  appointed  to  sustain,  so  that  the  unguents 
may  not  be  expelled.     We  would  also  remark  that  all  the 
experiments  in  consideration  were  made  under  pressures 
more  or  less  considerable,  and  their  results  should  only 
be  applied  to  analogous  cases.     In  fact,  we  may  conceive 
that  if  the  pressures  were  so  great,  in  respect  to  the  sur- 
faces, as  to  occasion  a  marked  defacement,  the  state  of  the 
surfaces,  and  consequently  the  friction,  would  vary ;  or 
that,  on  the  contrary,  if  the  surfaces  were  great,  and  the 
pressures  very  slight,  the  viscosity  of  the  unguents,  usu- 
ally disregarded,  might  then  exert  a  sensible  influence. 

"We  would  observe,  that  in  general,  and  especially  for 
metals,  pure  water  is  a  bad  unguent,  and  often  increases, 
rather  than  diminishes  the  friction. 

236.  Adhesion  of  mortar  and  solidified  cements. — But, 
for  mortars  which  have  set  and  acquired  a  proper  degree 
of  dryness,  there  exists  a  different  condition  of  tilings. 
Adhesion  and  cohesion  take  the  place  of  friction,  and  the 
resistance  to  separation  becomes  sensibly  proportional  to 
the  extent  of  the  surface  of  contact,  and  independent  of 
the  pressure  exerted,  either  at  the  moment  of  rest,  or  that 
of  separation. 


FRICTION.  283 

For  limestones  bedded  with  mortar  of  hydraulic  lime 
of  Metz,  the  resistance  is  about  2112lbs-  per  square  foot  of 
surface.  With  other  limes,  undoubtedly  common,  M. 
Boistard,  Engineer  of  Koads  and  Bridges,  has  found 
14:26lbs-  With  plaster,  the  resistance  seems  to  follow  the 
same  law,  but  it  varies  considerably  with  the  instant  of 
the  setting  of  the  plaster,  which  seems  to  exert  a  great 
influence  upon  the  cohesion. 

237.  Observation  upon  the  introduction  of  friction  and 
cohesion  in  calculations  upon  the  stability  of  construc- 
tions.— Finally,  we  would  remark  that  friction  cannot,  in 
the  case  of  beddings  in  mortar,  or  in  plaster,  show  itself 
until  the  cohesion  or  adhesion  is  overcome,  and  that  con- 
sequently these  two  resistances  cannot  coexist.     In  cal- 
culations upon  the  stability  of  structures,  we  should  only 
reckon  upon  one  of  these,  and  that  the  weakest. 

238.  Experiments  upon  friction  during  a  shock. — From 
the  general  opinions  which  we  have  published  upon  the 
action  of  forces,  and  the  efforts  of  compression  developed 
during  a  shock,  and  from  the  verification  in  Nos.  66  and 
67,  of  the  consequences  derived  from  them,  we  are  justi- 
fied in  concluding  that  the  efforts  produced  during  the 
shock  occasion  frictions,  which  follow  exactly  the  usual 
laws  of  friction.     It  is  moreover  expressly  admitted  by 
the  illustrious  Poisson,  who,  in  the  second  edition  of  his 
"  Traite  de  Me'canique,"  No.  475,  expresses  himself  in 
these  terms :  "  Though  no  observations  have  been  made 
upon  the  intensity  of  friction  during  a  shock,  we  may 
suppose,  by  induction,  that  it  follows  the  general  laws  of 
friction  of  bodies  subjected  to  pressures,  since  percussion 
is  only  a  pressure  of  very  great  intensity  exerted  during  a 
very  short  time." 

To  verify  by  direct  observation  the  correctness  of  this 
supposition,  and  at  the  express  invitation  of  M.  Poisson,  I 
undertook  many  series  of  experiments,  choosing  for  that 
purpose  the  case  of  strips  of  cast-iron  sliding  upon  bars 


284 


FKICTION. 


FIG.  99. 


of  cast-iron  spread  with  lard,  since  this  had  been  the  sub- 
ject of  careful  study  in  my  preceding  experiments,  and 
is  the  case  which  most  frequently  occurs  in  practice. 

239.  Description  of  the  apparatus  employed  in  the  ex- 
periments.— The  apparatus  which  I  employed  differs  from 

that  described  in  No.  225, 
only  in  the  following  dis- 
position necessary  for  sus- 
pending to  the  sled,  at  a 
desired  height,  the  body  de- 
signed to  produce  the  shock, 
and  allowed  to  fall  at  will 
during  the  motion. 

Upon  the  sides  of  the 
box  of  the  sled  are  raised 
two  frames  of  firm  uprights 
ab  and  #'£',  pierced  with 
holes  at  intervals  of  .16f%  through  which  pass  two  iron 
pins  ;  upon  these  pins  rests  a  movable  crosspiece  cd  of 
oak.  By  raising  and  lowering  the  pins,  the  height  of  the 
crosspiece  cd  above  the  sled  may  be  varied  at  will.  A 
screw  e  and  nut  passes  freely  across  a  hole  cut  in  the  mid- 
dle of  the  crosspiece,  and  bears  a  plyer  with  ring  legs, 
upon  which  is  suspended  a  shell  to  give  the  shock.  The 
two  legs  of  the  plyers  are  bound  with  strips  of  wick  with 
quick  match,  holding  them  shut.  By  means  of  the  screw 
e  the  height  of  the  shell  above  the  surface  shocked  can 
be  exactly  regulated. 

We  may  easily  conceive  from  this  description,  the  box 
and  uprights  being  firmly  fastened  to  the  sled,  that  the 
whole  system  partakes  of  a  common  motion,  and  that  if 
at  any  instant  of  its  course,  the  shell  falls  upon  the  sled, 
it  falls  there  with  a  vertical  velocity  due  to  the  height  of 
the  fall,  and  with  a  horizontal  velocity  which,  as  we  shall 
see  hereafter,  was  sensibly  the  same  as  that  of  the  sled. 
By  means  of  the  ligature  of  the  legs  of  the  plyers  we  ac- 
complish the  fall  of  the  shell,  without  any  external  con- 


FRICTION.  285 

cussion  or  disturbance.  For  this  purpose,  a  man  sets  fire 
to  the  match,  and  gives  the  signal  for  the  starting  of  the 
sled ;  combustion  is  communicated  to  the  upper  part 
which  keeps  the  plyers  closed  ;  these  open  suddenly  and 
let  loose  the  shell,  without  any  possibility  of  disturbing 
the  common  motion  of  the  system  of  the  two  bodies. 

240.  General  circumstances  of  the  experiments. — The 
experiments  were  made  in  impressing  the  sled,  sometimes 
with  a  uniform,  and  sometimes  with  an  accelerated  mo- 
tion.    The  first  of  these  motions  was  obtained  at  will,  by 
giving  to  the  descending  box  a  weight  just  sufficient  to 
overcome  the  friction,  and  in  suspending  under  this  box  a 
shell  of  110ib3-  weight,  which  only  descended  1.64"-  when 
its  action  ceased.     As  for  the  accelerated  motion,  it  was 
produced  whenever  the  motive  weight  surpassed  the  fric- 
tion.    The  law  of  these  motions  was,  moreover,  deter- 
mined, in  each  case,  by  means  of  curves  traced  by  the 
style  of  the  chronometric  apparatus. 

241.  General  examination  of  what  occurred  in  these 
experiments. — We  can  readily  appreciate  the  mode  of  ac- 
tion during  the  experiments.     We  take,  for  example,  a 
case  where  the  system  of  the  sledge  and  the  shell  sus- 
pended above  it,  is  impressed  with  a  uniform  motion. 
At  the  instant  when  the  combustion  of  the  wick  allows 
the  legs  of  the  plyers  to  separate,  the  shell  is  free,  and 
falls ;  during  its  fall,  until  the  moment  it  reaches  the 
sledge,  the  latter  being  freed  from  the  weight  of  the  shell, 
acquires  an  amount  of  motion  precisely  equal  to  what 
would  be  consumed  by  the  friction  due  to  this  weight. 
The  horizontal  velocity  of  the  sledge,  at  the  instant  of  the 
shock,  is  then  a  little  greater  than  that  of  the  shell.     After 
this  the  forces  of  compression  developed  by  the  shock 
produce  a  friction  variable  as  themselves,  at  each  instant, 
which  consumes  a  certain  quantity  of  motion ;  so  that  the 
sledge,  whose  progress  was  accelerated  during  the  fall  of  the 
shell,  is  afterwards  retarded  during  the  action  of  the  shock. 


286  FRICTION. 

242.  FormuloB  employed  in  calculating  the  results  of 
the  experiments.  —  As  it  is  desirable  to  prove  whether  the 
friction  remained  proportional  to  the  variable  pressures 
produced  during  the  short  intervals  of  the  phenomena, 
we  proceed  to  give  some  formulae  relative  to  this  hypothe- 
sis, which  we  will  hereafter  compare  with  the  results  of 
experiment.  We  consider  first  the  case  of  uniform  mo- 
tion, and  call 

Q  the  weight  of  the  sledge3  and  the  suspending  appa- 
ratus of  the  shell  ; 

q  the  weight  of  the  shell  producing  the  shock  ; 

f  the  ratio  of  friction  to  the  pressure  for  the  surfaces 
in  contact; 

h  the  height  of  fall  of  the  shell  above  the  sledge  ; 

U  the  velocity  due  to  this  height  ; 

T  the  time  of  the  fall  ; 

Y  the  horizontal  velocity  of  the  sledge  and  shell  at  the 
instant  when  the  latter  is  let  loose  by  the  plyers  ; 

V  the  velocity  of  the  body  after  the  shock  ; 


At  the  instant  when  the  shell  is  freed,  the  quantity  of 
motion  of  the  system  is 


ff 

The  weight  of  the  shell,  when  connected  with  the 
sledge,  produces  a  friction  fq  which,  in  each  element  of 
time  t,  consumes  a  quantity  of  motion  fqt^  and  which, 
during  the  time  of  the  fall,  would  consume  the  quantity  /^T. 

But  since,  on  the  other  hand,  the  shell  ceases  to  press 
upon  the  sledge  during  this  time,  it  follows  that  the  quan- 
tity of  motion  gained  by  the  system  by  reason  of  this 
diminution  of  pressure,  is  precisely  fqT. 

At  the  instant  when  the  shell  reaches  the  sledge,  the 
quantity  of  motion  possessed  by  the  system  is  then 


FRICTION.  287 

From  this  instant,  and  during  the  whole  period  of  the 
shock,  the  shell  loses,  in  each  element  of  time,  an  element 

of  velocity,  and  consequently  a  quantity  of  motion  -u, 

whence  results  a  force  of  compression  -  x  -,  producing  a 

9    t 

friction1^  x-.     This  friction  consumes,  in  an  element  of 

9     t 
time  a  quantity  of  motion  -S  .  ^,  and  when   all  relative 

'  •  '•     . 

motion  in  a  vertical  direction  is  destroyed,  the  friction 

due  to  the  forces  of  compression  has  finally  consumed  a 

quantity  of  motion  equal  to  —  U. 

y 

Consequently,  when   the  shock  has  terminated,  we 
should  have  between  the  quantities  the  relation 


or 


Now,  the  shell  falling  with  a  uniformly  accelerated 
motion  by  virtue  of  gravity,  we  have,  evidently,  TJ=^T, 
whence  it  follows  that  Y—  V  ;  that  is  to  say,  that  in  our 
apparatus  the  quantity  of  motion  destroyed,  by  the  fric- 
tion resulting  from  the  forces  of  compression,  must  be 
precisely  equal  to  that  which  it  gains  during  the  fall  of 
the  shell. 

These  two  effects  are  successive,  but  take  place  in  a 
short  interval  of  time,  and  therefore  occasion  in  the  curve 
of  motion  undulations  in  opposite  directions,  which  do 
not  affect  the  general  law,  and  are  scarcely  appreciable, 
either  in  the  draughted  curve  or  that  made  from  the  ab- 
stract of  the  table. 

243.  The  acceleration  of  the  motion  of  the  sledge  dur- 
ing the  fall  of  the  shell  may  ~be  neglected.  —  It  is  easy  to  be 


288  FRICTION. 

assured  a  priori,  that  the  acceleration  of  the  velocity  of 
the  sledge  during  the  fall  of  the  shell,  was  always  very 
small  in  our  experiments,  though  the  height  of  the  fall 
has  reached  1.97ft-  We  observe,  then,  from  what  has 
just  been  said,  that  calling  Vx  the  horizontal  velocity  of 
the  sledge,  at  the  moment  when  the  shell  reaches  it,  we 
shall  have 


whence 

VTr 

Making,  for  example, 

2=11 
and 

U=13.80ft-,     Q=590.68lb%    /=0.071, 

which  answers  to  one  of  the  most  intense  shocks  produced 
during  the  experiments,  we  find 

¥,—¥=0.1829"- 

Now,  the  shock  of  the  shell  in  the  horizontal  direction 
taking  place  only  in  virtue  of  this  difference  in  velocity, 
we  see  that  its  effect  upon  the  general  motion  should  be 
quite  insensible,  and  we  may,  as  we  have  done  in  the  pre- 
ceding calculation,  neglect  its  influence  upon  the  general 
motion  of  the  sledge. 

244.  Case  where  the  motion  of  the  sledge  is  acceler- 
ated. —  The  preceding  reasoning  applies  to  the  case  where 
the  system  of  the  shell  and  of  the  sledge  is  impressed 
with  an  accelerated  motion,  and  it  follows  that  if,  as  we 
have  admitted,  the  friction  during  the  shock  remains  pro- 
portional to  the  pressure,  the  general  law  of  motion  in  our 
apparatus  cannot  be  affected  ;  or,  in  other  words,  that  if, 


FRICTION.  289 

before  the  fall  of  the  shell,  the  motion  is  uniform  or  accel- 
erated, according  to' a  certain  law,  it  will  still  be  so  after 
the  shock,  according  to  the  same  law.  The  only  disturb- 
ance which  will  result  will  be  sometimes  manifested  by 
undulations,  which,  in  most  cases,  would  hardly  be  appre- 
ciable. Moreover,  the  hardness  or  compressibility  of  the 
body  in  contact  should  not  have  any  influence  upon  the 
result,  and  in  causing  the  shell  to  fall  upon  the  beech- 
wood  joists  composing  the  sledge,  or  upon  a  mass  of  soft 
loam  placed  upon  it,  we  should,  for  circumstances  other- 
wise similar,  find  the  same  law  of  motion,  which  should 
be  the  same  as  though  there  had  been  no  shock. 

245.  Results  of  experiments. — It  remains  now  for  us 
to  compare  the  results  of  the  formulae  with  those  of  ex- 
periments which  have  been  made,  some  when  the  sledge 
was  impressed  with  a  uniform  motion,  and  some  when 
the  motion  was  accelerated.  In  these  experiments,  we 
have  varied  the  weight  of  the  shells  imparting  the  shock 
from  26.43lbs-  to  110lb%  or  nearly  1  to  4 ;  the  ratio  of  the 
weight  of  the  body  imparting  the  shock  to  that  of  the 
body  shocked,  from  TV  to  £,  and  the  height  of  the  fall 
from  0.32Sft-  to  2.29f%  or  from  1  to  7.  The  shock  was 
produced  upon  wood,  and  upon  loam  placed  upon  the 
sledge.  If,  then,  the  laws  which  we  have  admitted  in 
the  preceding  formulas  are  verified  by  experiments  within 
such  extended  limits,  we  may,  I  think,  conclude  that  they 
subsist  for  pressures  developed  during  the  shock,  as  well 
as  for  others  without  shocks. 

I  shall  only  publish  here  the  experiments  made  in  the 
case  of  uniform  motion,  where  the  shock  was  made  upon 
wood  and  loam.  Experiments  made  with  motive  weights 
which  produced  an  accelerated  motion,  have  led  to  simi- 
lar consequences :  the  acceleration  produced  having  al- 
ways been  sensibly  the  same  in  the  case  of  a  shock  as  in 
those  where  there  was  none. 
19 


290 


FRICTION. 


•*S          cJ 

•I    * 
I    % 


V        ? 
V        o 

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$ 

.1 
§ 


03 

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s 


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I 

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<M'  N  oq  <M'  cq"  d  «  <N  <N  <N  <N  IM"  c<j  <N  <^  <ri  co  <M'  eo 


oooooooooooooqqqqoo 
d  d  d  d  d  d  d 

•*  <M<Mcocococoeococoeocooooooo 

c<j  oooooooooqooooo^»»o^SS§ 

OS       >0  >0  00*  CO  CO  CO  CO  00  00  00  00  <M'  <N  (M' 

co          cococococococococococo 
^   „ 

rH  OOQOOOOOOOCOCOOOOOOO^'tl^ 

CO  *>  l>  00  00  00  00  00  00  00  00  00  O5  OS  O5 

.fc'S  5       ""          ^         J5  co^®  °*  °"  ®  o>c*  ®  ®  "*'  "*  **' 

I 

dddddr-?r-Jdddi-irH»-HTHc<ic<id 
2 

06"  "*-*cococococococococoaooo 

JH  oqcococococococococooooo 

fl~~~ — 

I 

(MiMcoeocococoeocoeocot^b-t- 

•^'H/lrHrHrHiHr-ir- IrHi— lrH<N<NW 

*-    " 

"8 


ii 
ss's 


FRICTION. 


291 


•§ 


a 


Is 


%  a 

«  bog 

p 

a  1 


ight  of  f 
of  the 
phere  h. 


^J  00  GO  CD        lO  ^  1C  00  GO  ls» 

'•-'OSCS  OS         OS  OS  OS  OS  O5  OS 

OO*  r-I         (N  <?i  <M*  O  O  r-J 


S 


CNj 

si 


292  FRICTION. 

"We  see  by  these  tables  that  the  velocity  of  uniform 
motion  has  been  the  same  in  the  experiments  made  with 
the  shocks  as  in  those  without  them,  whatever  may  have 
been  the  height  of  the  fall.  This  velocity,  in  all  cases, 
has  depended  solely  upon  the  load  or  total  pressure  of  the 
motive  weight,  and  the  state  of  the  surfaces. 

An  examination  of  the  curves  of  motion  shows,  from 
the  vibrations  produced  by  the  shock  throughout  the  ap- 
paratus— which  are  felt  even  at  the  style — in  what  place 
the  shock  was  produced,  and  whether  it  occurred  in  the 
period  of  its  course,  when  the  motion  had  become  uni- 
form, or  in  that  when  it  was  accelerated,  the  draughted 
curve  and  the  abstract  of  the  tables  afford  but  slight  undu- 
lations, and  the  motion  remains  or  becomes  uniform  with 
the  same  velocity. 

Finally,  these  experiments  show  that  in  the  shock  the 
frictions  due  to  the  pressures  developed,  are  still  propor- 
tional to  these  pressures,  and  independent  of  the  velocity. 

246.  The  transmission  of  'motion  l>y  means  of  lelts. — 
The  theory  of  the  transmission  of  motion  by  means  of  cords 
or  endless  belts  is  founded  upon  two  theories.     The  first, 
that  of  M.  Prony,  relative  to  the  sliding  of  a  cord  or  belt 
upon  the  surface  of  a  drum ;  the  second,  due  to  M.  Pon- 
celet,  refers  to  the  variation  of  tension  in  the  two  parts  of 
the  cord  or  belt  employed  in  these  transmissions.     I  pro- 
pose to  prove,  by  special  experiments,  the  consequences 
of  these  two  theorems,  and  proceed  to  give  a  succinct 
account  of  the  results  of  these  researches. 

247.  The  slipping  of  cords  or  lelts  upon  cylinders. — 
In  explaining  the  first  of  these  theorems,  let  us  consider 
a  cord  or  belt  enveloping  a  portion  of  the  surface  of  a 
cylinder,  and  acted  upon  at  one  end  by  a  power  P,  and 
upon  the  other  by  a  resistance  Q.     It  is  clear,  that  to  pro- 
duce the  slipping  of  the  cord,  the  power  P  should  be 
equal  to  the  resistance   Q,  increased  by  the  resistance 


FEICTION.  293 

opposed  by  the  friction  of  the  cord  upon  the  surface  of  the 
cylinder.  Let  us  seek  to  determine  this  friction. 

For  this  purpose,  we  consider  the  two  consecutive  ele- 
ments ah  and  be  of  the  cord,  and  call : 

T  the  tension  of  the  cord  in  the  element  ab. 

Tr  the  tension  of  the  cord  in  the  element  5<?. 

It  is  evident  that  the  tension  T7  exceeds  the  tension  T 
by  an  infinitely  small  quantity  £,  which  is  precisely  the 
measure  of  the  resistance  opposed  by  the  friction;  we 
have  then 


and  passing  from  one  element  to  the  other,  from  the  point 
n  of  contact  of  the  direction  nP,  where  T=P  to  the  point 
in  of  contact  of  the  direction  mQ,  where  T=Q,  the  sum 
of  all  the  increments  of  tension  produced  by  the  friction 
at  the  moment  of  slipping,  will  give  the  total  tension. 

The  friction  or  elementary  in- 
crease of  tension  t,  from  the  ele- 
ment ab  to  the  element  fo,  is 
produced  by  the  pressure  result- 
ing from  the  component  of  tension 
T',  normal  to  the  surface,  which 
is  T  sin  «,  calling  a  the  infinitely 
small  angle  at  the  intersection  of  Fia  m 

the  two  elements  ab  and  fo,  or  simply  Ta,  since  T  differs 
by  an  infinitely  small  quantity  from  T7,  and  the  sine  a 
from  a  ;  we  have  then 

<=/.T«=T/|, 

/  being  the  ratio  of  the  friction  to  the  pressure. 

The  sum  of  all  these  increments  of  tension,  taken  from 
the  point  m  where  T=Q  to  the  point  n,  where  T=P, 
leads,  according  to  the  rules  of  analysis,  to  the  formula 

log  P=  log  Q  +  0.434/|,  or  P=Q .  2.7184 
S  being  the  total  length  embraced  by  the  cord. 


294:  FRICTION. 

"We  see  by  this  expression  that  the  tension  of  the  mo- 
tive power  increases  from  P=Q,  answering  to  S=0, 

g 
proportionally  to  the  opening  of  the  angle  =  ,    embraced 

by  the  cord,  and  not  to  the  absolute  extent  of  the  arc  ; 
which  shows,  from  theoretic  considerations,  that  for  an 
increase  of  the  friction  of  slipping  of  cords  or  belts,  it  is 
not  essential  to  enlarge  the  diameter  of  the  cylinder,  but 
that  the  proportional  part  of  the  circumference  to  be 
enclosed  should  be  increased. 

The  preceding  formula  relates  to  the  case  where  the 
power  P  is  to  overcome  the  resistance  Q,  and  conse- 
quently besides  this  to  surmount  the  friction  of  the  cord 
or  belt  upon  the  drum.  When,  however,  as  is  frequently 
the  case,  the  force  P  is  to  yield  to  the  force  or  weight  Q, 
for  moderating  its  action,  or  resisting  it  altogether,  as,  for 
example,  in  the  lowering  of  goods,  the  friction  acts  in 
favor  of  the  force  P,  and  we  have 


log  P-  log  Q-0.484/.     ,  or  P= 


.. 


Such  are  the  relations  which  theory  indicates  between 
the  forces  P  and  Q,  the  arc  of  contact,  the  radius  of  the 
drum,  and  the  coefficient  of  friction.  It  remains  to  de- 
termine by  experiment  the  correctness  of  these  relations. 

248.  Experiments  upon  the  slipping  of  cords  and  of 
belts  upon  the  surface  of  wooden  drums,  and  of  cast-iron 
pulleys.  —  For  this  purpose  I  made  use  of  three  wooden 
drums,  with  diameters  of  2.741ft-,  1.338f%  and  0.328"-, 
placing  them  horizontally  in  a  fixed  position,  so  that  they 
could  not  turn,  and  over  them  was  passed  a  belt  of  black 
curried  leather,  nearly  new,  but  having  acquired  a  certain 
pliability  from  previous  use.  Its  breadth  was  0.164:ft;,  and 
thickness  0.1T3ft-  ;  its  rigidity  seemed  so  feeble  that  we 


FRICTION.  295 

were  justified  in  neglecting  it  in  its  ratio  to  the  friction 
of  slipping  upon  the  surface  of  the  drum. 

The  two  strips  of  the  belt  hung  vertically  in  equal 
portions  on  each  side  of  the  drum,  and  to  each  of  them 
was  attached  a  scale  to  receive  the  weights.  The  belt 
weighed  5.06lb%  each  scale  0.5lbs> ;  consequently,  the 
weight  of  each  strip,  of  equal  length,  was,  with  its  plate, 
3.03lbs>  The  arc  embraced  was  equal  to  the  semi-circum- 
ference. At  first,  equal  weights  were  put  in  the  scales, 
then  gradually  was  added  to  one  of  them  the  weights 
necessary  to  make  the  belt  slide  upon  the  drum. 

We  see  from  this,  that  the  tension  Q  of  the  ascending 
strip  was  equal  to  3.03lbs-  plus  the  weight  contained  in  the 
corresponding  scale,  and  that  the  tension  P  of  the  de- 
scending strip  was  equal  to  Q  increased  by  the  weight 
added,  over  and  above  the  primitive  load. 

This  established,  the  preceding  formula  becomes 

log  P=  log  Q+0.434/.  ^  =  log  Q+0.434/ x  3.1416, 

whence  we  deduce 

felog  P~  log  Q^log  P-log  Q 
J     0.434  x  3.1416  1.363 

By  introducing  in  this  formula  the  values  of  P  and  Q 
furnished  by  experiments,  we  are  enabled  to  calculate 
the  different  values  of  the  ratio  f  of  the  friction  to  the 
pressure,  and  to  be  assured  that  they  confirm  the  theoretic 
consequences  which  we  have  unfolded. 

249.  Results  of  'experiments.— The  two  following  tables 
contain  the  results  of  the  experiments : 


FRICTION. 


TABLE. 

Experiments  upon  the  friction  of  belts  upon  wood  drums. 


Width  of 
belt. 

Condition  of 
the  belt. 

Diameter  of 
drum. 

Length 
of  arc 
embraced. 

Tension  of  the  part. 

Ratio  of 
'riction 
to  pres- 
sure f. 

rising 
Q. 

falling 
P. 

ft. 

ft. 

ft. 

Ibs. 

Ibs. 

1 

( 

{ 

14.060 

66.992 

0.497 

14.060 

64.786 

0.486 

Dry 

\ 

14.060 

64.786 

0.492 

0.164  - 

somewhat 
oily. 

2.741  - 

4.306  -j 

36.114 
36.114 

167.341 
153.336 

0.488 
0.460 

36.114 

151.461 

0.458 

25.087 

111.102 

0.473 

, 

. 

I 

25.087 

95.603 

0.426 

Mean 

0.472 

f 

14.060 

63.683 

0.472 

14.060 

69.197 

0.458 

2.099  \ 

14.060 

63.242 

0.507 

I 

36.114 

140.875 

0.479 

I 

36.114 

140.875 

0.433 

Mean  

0.462 

j- 

1 

• 

14.060 

73.608 

0.526 

14.060 

75.813 

0.541 

Dry 

25.087 

91.252 

0.411 

0.164  - 

somewhat  \ 

0.328  - 

0.514  - 

25.087 

98.975 

0.438 

oily. 

25.087 

94.560 

0.422 

36.114 

161.827 

0.477 

. 

J 

. 

36.114 

168.576 

0,490 

Mean  

0.472 

1' 

r 

11.911 

71.458 

0.570 

11.911 

72.560 

0.575 

0.091  - 

4.306  - 

22.938 
22.938 

114.465 
104.541 

0.512 
0.483 

33.965 

137.622 

0.446 

. 

33.965 

136.519 

0.443 

Mean.... 
General  mean.... 


,.  .504 
,.0.477 


FRICTION. 


297 


I 
I 

3 


- . 


^ 


£1 
*i,s 

|3 


mbr 

S. 


1* 


"o 
-«*J 

II 


21 

^i° 

l§-d 
Jiul 

af5j 


3^2, 

le 


II 
111 

-23 


I!! 


IO1C4CMCO     I    C1-*         <N  CO  <M  CO  < 

>  d  o  d  o  I  o     o  d  o  o  < 


Q$         TH  CO  O  CO  O  1O 

o    oo*cj2c>GJ|<5> 


£  C5  O  T*  ••*  OS  C'l 
oS  CO  CO  «0  00  CO 


s 


298  FRICTION. 

We  see  by  the  results  of  these  experiments,  in  which 
the  arc  of  contact  varied  in  the  ratio  of  8.3  to  1  nearly, 
and  where  the  tension  has  reached  very  nearly  the  limits 
assigned  to  the  belts  of  machinery,  that  the  value  of  the 
ratio  /  of  friction  to  the  pressure,  remained  very  nearly 
constant. 

The  three  first  series  of  the  first  table  fully  confirm  the 
theoretic  considerations.  The  fourth  series  relates  to  a 
belt  quite  new,  and  very  stiff,  and  to  this  we  attribute  the 
small  increase  presented  by  it  in  the  mean  value.  This 
belt  having,  moreover,  only  a  width  of  .091f%  or  about 
the  half  of  the  preceding,  we  see  that  this  last  series  con- 
firms, as  to  belts,  the  law  of  the  independence  of  sur- 
faces. 

In  the  experiments  of  the  second  table,  the  extent  of 
arc  embraced  varied  in  the  ratio  of  6  to  1,  the  breadth  of 
the  belt  pressed  against  the  pulley  in  that  of  2  to  1,  the 
tension  from  1  to  3  and  from  1  to  6,  and  still  the  value 
of  the  ratio  f,  of  friction  to  the  pressure  remained  sensibly 
constant,  and  equal  in  the  mean,  for  the  dry  belt  and  dry 
pulleys 

/=0.282. 

When  the  pulley  was  moistened  with  water  we  had 


250.  Conclusions.  —  In  considering  the  results  of 
these  two  series  of  experiments  upon  the  friction  of  belts 
upon  wooden  drums  and  cast-iron  pulleys,  we  see  that  we 
are  justified  in  admitting  that  the  ratio  of  the  resistance 
to  the  pressure  is  : 

1st.  Independent  of  the  width  of  the  belt  and  of  the 
developed  length  of  the  arc  embraced,  or  of  the  diameters 
of  the  drums,  or,  what  amounts  to  the  same,  are  inde- 
pendent of  the  surface  of  contact. 

2d.  Proportional  to  the  angle  subtended  by  the  belt  at 
the  surface  of  the  drum. 


FRICTION.  299 

3d.  Proportional  to  the  logarithm  of  the  ratio  of  the 
tension  of  the  strips,  and  expressed  by  the  formula 


- 

J~ 


1.363 

251.  Experiments  upon  the  variation  of  the  tension  of 
endless  cords  or  l>elts  used  in  transmitting  motion.  —  We 
pass  now  to  an  experimental  proof  of  the  theory  given  by 
M.  Poncelet,  upon  the  transmission  of  motion  by  endless 
cords  or  belts,  and  will  first  give  a  description  of  its 
nature. 

When  a  cord  or  belt  surrounds  two 
pulleys  or  drums,  between  which  it  is  de- 
signed to  maintain  a  conjoint  motion,  care 
is  taken  to  give  it  a  sufficient  tension, 
which  is  usually  determined  by  trial,  but 
which  it  would  be  best  to  calculate,  as  we 
shall  see  hereafter.  The  primitive  tension 
is,  at  the  commencement,  the  same  for 
both  parts  of  the  belt,  and  this  equality 
established  in  repose,  is  only  destroyed  by 
the  friction  of  the  axles,  which  may  act  in 
either  direction  according  to  that  of  the  FIG.  101. 
motion  of  the  pulleys. 

Let  us  examine  how  this  motion  is  transmitted  in  such 
a  system.  Let 

C  be  the  motive  drum  ; 

C'  the  driven  drum  ; 

Tj  the  primitive  tension  common  to  the  parts  AA'  and 
BB'  of  the  belt,  from  the  moment  when  the  drum  C  be- 
gins to  turn  until  it  commences  to  turn  the  drum  C'. 

The  point  A  of  primitive  contact  of  the  part  AA'  ad- 
vances, in  separating  from  the  point  A',  in  the  direction 
of  the  arrow,  the  strip  A  A'  is  stretched,  and  its  tension 
increased  by  a  quantity  proportional  to  this  elongation, 


300  FRICTION. 

according  to  a  general  law  proved  by  experiment  upon 
traction.*  At  the  same  time,  the  point  B  of  contact  of 
the  part  BB7,  approaches  by  the  same  quantity  towards 
the  point  B7,  so  that  the  portion  BB7  is  diminished  by  a 
quantity  equal  to  the  increase  of  that  of  AA7.  If,  then, 
we  call 

T  the  tension  of  the  driving  portion  AA7,  at  the  in- 
stant of  its  being  put  in  motion, 

T'  the  tension  of  the  driven  part  BB7, 

t  the  quantity  by  which  the  primitive  tension  Ta  is 
increased  in  the  portion  A  A7,  and  diminished  in  the  part 
BB',  we  shall  have 

T=Tt  +  t,  and  T/=T1-«, 

and  consequently 


Then,  at  any  instant,  the  sum  of  the  two  tensions  T 
and  T7  is  constant  and  double  the  primitive  tension. 

Now  it  is  evident,  that  in  respect  to  the  driven,  drum 
C7  the  motive  power  is  the  tension  T,  and  that  the  tension 
T7  acts  as  a  resistance  with  the  same  lever  arm,  so  that 
the  motion  is  only  produced  and  maintained  by  the  excess 
T  —  T7  of  the  first  over  the  second  of  these  tensions. 

If  the  machine  is,  for  example,  designed  to  raise  a 
weight  Q  acting  at  the  circumference  of  an  axle  with  a 
radius  R7,  it  is  easy  to  see,  according  to  the  theory  of 
moments,  that  at  any  instant  of  a  uniform  motion  of  the 
machine,  we  must  have  the  relation 


!N"  being  the  pressure  upon  the  journals,  and  r  their 
radius. 


*  See  Lessons  -upon  "  Resistance  des  Materiaux. 


FEICTION.  301 

The  pressure  is  easily  determined;  for  calling 
a  the  angle  formed  by  the  directions  AA'  and  BB'  of 
the  belts  with  the  line  of  the  centres  CC'. 
M  the  weight  of  the  drum. 
We  see  immediately  that 

N=  V[M+Q+(T— T)  sin  af+(T+T)  cos'a, 

an  expression  which,  according  to  the  algebraic  theorem 
of  M.  Poncelet,  cited  in  No.  227,  has  for  its  value  to  ~ 
nearly,  when  the  first  term  under  the  radical  is  greater 
than  the  second, 

~J$T=0.96  [M+Q+(T-TO  sin  a]+0.4  (T+T)  cos  a. 

This  value"  of  K  being  introduced  into  the  formula  for 
equality  of  moments,  we  have  a  relation  containing  only 
the  values  of  the  resistance  Q  and  of  the  tensions.  But 
as  it  may  be  somewhat  complicated  for  application,  ob- 
serving that  in  most  cases  the  influence  of  the  tensions  T  and 
T'  upon  the  frictions,  will  be  so  small  that  it  may  be 
neglected,  at  least  in  a  first  approximation,  we  proceed 
as  follows : 

First,  neglecting  the  influence  of  the  tensions  upon 
the  friction,  we  have  simply,  in  the  actual  case, 


and  consequently 

(T-T')K=QE'+/(M+Q)r, 
whence  we  deduce 

^=Q. 


B 

which  furnishes  a  first  value  for  the  difference  of  tensions, 
which  is  the  motive  power  of  the  apparatus. 

But  this  is  not  sufficient  to  make  known  these  ten- 


302 


FRICTION. 


sions,  and  it  is  necessary  to  determine  the  primitive  ten- 
sion T',  so  that  in  no  case  the  belt  may  slip. 

According  to  the  theory  of  M.  Prony,  we  have,  at  the 
instant  of  slipping,  between  the  tension  T  and  T'  the  re- 
lation 


The  number  K  being  a  quantity  depending  upon  the 
nature  and  condition  of  the  surfaces  of  contact,  as  well  as 

Q 

upon  the  angle  ^  embraced  by  the  belts  upon  the  drum 

C'.  These  quantities  are  known,  and  we  may  in  each  case 
calculate  the  value  of  K  by  this  formula,  or  take  it  from 
the  following  table,  which  answers  to  nearly  all  the  cases 
in  practice  : 


T?ofi/\  r\f 

VALUE    OP    THE    RATIO     K. 

JttltlO  01 

the  arc 
embraced 

New  belts 

Belts  in  usual  con- 
dition. 

Moistened 

X/-klfo 

Cords  upon  wooden  drums  or 
axles. 

to  the 

DGltS 

circumfer- 
ence. 

wooden 
drums. 

upon 
wooden 
drums. 

upon 
cast-iron 
pulleys. 

upon 
cast-iron 
pulleys. 

Bough. 

Smooth. 

0.20 

1.87 

1.80 

1.42 

1.61 

1.87 

1.51 

0.30 

2.57 

2.43 

1.69 

2.05 

2.57 

L86 

0.40 

3.51 

3.26 

2.02 

2.60 

3.51 

2.29 

0.50 

4.81 

4.38 

2.41 

3.30 

4.81 

2.82 

0.60 

6.59 

5.88 

2.87 

4.19 

6.58 

3.47 

0.70 

9.00 

7.90 

3.43 

5.32 

9.01 

4.27 

0.80 

12.34 

10.62 

4.09 

6.75 

12.34 

5.25 

0.90 

16.90 

14.27 

4.87 

8.57 

16.90 

6.46 

1.00 

23.14 

19.16 

5.81 

10.89 

23.90 

7.95 

1.50 

111.31 

22.42 

2.00 

535.47 

63.23 

2.50 

2574.80 

178.52 

By  means  of  this  table,  we  shall  have  then  the  value 
of  T=KT',  and  consequently 

T-T/=(K— 1)T'=Q, 
Q  representing  the  greatest  value  which  the  difference 


FRICTION. 

of  tensions  should  attain,  to  overcome  the  useful  and  passive 
resistances. 

Froni  this  relation  we  may  derive  the  smallest  tension 
to  be  allowed  to  the  driven  portion  of  the  belt,  to  prevent 
its  slipping  :  we  thus  have 

/_    Q 


"We  should  increase  this  value  by  TV  at  least  to  free  it 
from  all  hazard  of  accidental  circumstances,  and  to  restore 
the  account  of  the  influence  of  the  tensions  upon  the  fric- 
tion, which  was  neglected.  This  established,  we  have 


and  consequently 

rp  __  •a-~r~  J-  -«•  -"-T~  -»• . 

All  the  circumstances  of  the  transmission  of  motion 
will  then  be  determined. 

If  these  first  values  of  T,  T',  and  Tx  are  not  considered 
as  sufficiently  correct,  we  may  obtain  a  nearer  approxi- 
mation by  introducing  them  in  the  value  of  the  pressure 
!N,  and  thus  deduce  a  more  exact  value  of  Q,  which  will 
serve  to  calculate  anew  T',  then  T  and  T,. 

252.  Experiments  upon  the  variations  of  tensions  of 
endless  "belts  employed  for  the  transmission  of  motion. — 
To  verify  by  experiment  the  exactness  of  these  considera- 
tions, I  placed  vertically  above  the  axis  of  a  hydraulic 
wheel,  and  of  a  pulley  mounted  upon  its  axle,  a  cylindri- 
cal oak  drum,  2.74:ft-  in  diameter,  and  whose  axis  was 
9.84:ft-  from  that  of  the  wheel.  Around  this  drum  A'B', 


304: 


FRICTION. 


102. 


and  the  pulley  AB,  was  passed  a  belt  which,  instead  of 
being  in  one  piece,  was  in  two  parts,  joined 
at  each  end  by  a  dynamometer  wit^i  a  plate 
and  style,  of  a  force  of  441lbs-  Moreover, 
these  dynamometers  were  easily  secured  in 
positions,  such  that  that  of  the  descending 
portion  of  the  belt  was  near  the  upper  drum, 
and  that  of  the  ascending  near  the  lower 
drum.  Thus  the  belt  could  be  moved  over 
a  space  of  6.56ft>,  without  the  risk  of  the  in- 
struments being  involved  with  the  drums. 
A  thread  wound  several  times  around 
the  circumference  of  one  of  the  grooves  of  the  plate  of 
each  of  the  dynamometers,  and  attached  by  the  other  end 
to  a  fixed  point,  caused  the  plate  to  turn  when  the  appa- 
ratus was  in  motion,  and  the  paper  with  which  the  plate 
was  covered  received  thus  the  trace  of  the  style  of  the 
dynamometer. 

The  belt  being  passed  over  the  two  drums,  the  ten- 
sions of  the  parts  were  varied  at  will,  in  either  direction, 
by  suspending  at  the  circumference  of  the  upper  drum  a 
plate  Q  charged  with  weights.  As  to  the  primitive  ten- 
sion, it  was  increased  by  bringing  nearer  together  the 
ends  of  the  belt,  or  in  diminishing  its  length  before  the 
experiment. 

The  apparatus  being  thus  prepared  for  observations, 
before  loading  the  plate  Q,  we  traced  the  circles  of  flexure 
of  each  of  the  dynamometers,  so  as  to  have  the  tensions 
of  the  belt  at  rest,  and  to  obtain  by  their  sum  the  double 
of  the  primitive  tension  Tr  We  may  conceive  that  these 
two  tensions  can  never  be  quite  equal,  but  that  is  not  im- 
portant, inasmuch  as  we  have  to  deal  only  with  their  sum. 
This  obtained,  we  load  the  plate  with  a  weight  which, 
being  suspended  upon  the  circumference  by  a  cord  of  a 
diameter  equal  to  the  thickness  of  the  belt,  has  the  same 
lever  arm  as  the  tensions.  That  part  of  the  belt  opposed 
to  this  weight  is  stretched,  and  the  part  on  the  same  side 


FRICTION. 


305 


is  slackened,  and  we  trace  the  new  curves  of  the  flexure 
of  the  dynamometers. 

For  the  same  primitive  tension  we  may  make  a  series 
of  experiments  up  to  the  motive  weight  under  the  action 
of  which  the  belt  slides  upon  either  drum. 

TABLE. 

Experiments  upon  the  variation  of  the  tensions  of  end- 
less belts  employed  in  transmitting  motion  by  pulleys  or 
drums. 


Number 
of  experi- 
ment. 

Weight 
suspended 
at  the 
circumfer- 
ence. 

Tension  of  the  part. 

Sum  of  the 
tensions 

T+T^. 

Remarks. 

Rising 
or 
stretched 
T. 

Descending 
or 
slackened 
T'. 

1 
2 
3 

Ibs. 
0.00 
44.61 
59.55 

Ibs. 
38.57 
60.07 
63.14 

Ibs. 
32.84 
12.84 
10.19 

Ibs. 
71.41 
72.91 
73.33 

The  belt  slipped. 

4 
5 
6 

7 
8 

0.00 
22.56 
44.61 
66.67 
97.54 

64.86 
75.09 
84.51 
97.96 
109.92 

57.41 
46.82 
36.26 
24.17 
20.76 

122.27 
121.90 
120.77 
122.13 
130.67 

\  The  dynamome- 
V  ters  moved  *' 
)  about  3.28  ft. 

9 
10 
11 

0.00 
55.64 
110.78 

73.73 
99.00 
117.77 

62.32 
41.53 
20.38 

136.05 
140.53 
138.15 

lid.,  id. 

12 
13 

0.00 
115.19 

66.91 
103.78 

57.94 
15.86 

124.85 
119.64 

The  belt  slipped. 

14 
15 
16 

0.00 
55.64 
110.78 

107.53 
130.05 

157.02 

98.91 
70.26 
47.57 

206.44 
200.31 
204.59 

17 
18 
19 

0.00 
110.78 
174.23 

97.24 
154.29 
170.67 

88-75 
40.78 
43.42 

185.99 
195.07 
214.09 

Do.     do. 

20 
21 

0.00 

88.72 

86.72 
134.84 

71.34 
44.17 

158.06 
179.01* 

*  Besides  the  load  Q 
there  was  suspended 
to  the  main  circum- 
ference of  the  floats 
of  the  wheel  at  6.05  ft. 
from  the  axis  a  weight 
of  22.56  Ibs.,  which 
broke  the  equilibrium 

20 


306  FRICTION. 

In  these  experiments,  facilities  were  afforded  for  allow- 
ing the  two  drums  to  turn  a  certain  amount,  under  the 
action  of  the  tensions,  so  that  we  could  realize  the  three 
cases  in  practice,  to  wit :  that  of  the  variation  of  tensions 
before  motion  was  produced,  that  of  the  variation  during 
motion,  and  finally,  that  of  the  slipping. 

The  belt  used  in  these  experiments  was  very  pliable, 
soft,  and  little  liable  to  be  polished  in  slipping.  In  cal- 
culating the  ratio  of  the  friction  to  the  pressure  for  this 
belt,  by  means  of  experiments  3,  13,  and  19,  we  find  re- 
spectively 

/=0.578,    /=0.596,    and   /=0.544, 

the  mean  being 

/=0.5Y3. 

253.  Remarks  upon  the  results  contained  in  the  pre- 
ceding table. — We  see  that  the  first  line  of  each  series 
corresponds  to  the  case  where  there  was  no  additional 
weight,  and  where  each  portion  of  the  belt  took  the  primi- 
tive pressure  corresponding  to  the  distance  apart  of  the 
axes.  As  the  weight  suspended  from  the  drum  was  in- 
creased, the  tension  of  one  of  the  strips  was  increased, 
and  that  of  the  other  was  diminished ;  but  so  that  their  sum 
remained  constant,  as  is  shown  by  the  fifth  column  of  the 
table. 

These  results,  which  completely  confirm  the  theory  of 
M.  Poncelet,  being  relative  to  tensions  whose  sum  reaches 
198lbs-  and  more,  where  the  greatest  rise  as  high  as  169lbs- 
and  the  smallest  fall  as  low  as  lllb%  comprise  nearly  all 
the  cases  in  practice,  and  show  that  this  theory  may,  with 
safety,  be  applied  to  the  calculation  of  transmission  of 
motion  by  belts. 

In  conclusion,  we  would  add,  that  belts  designed  for 
continuous  service  may  be  made  to  bear  a  tension  of 
0.551lbB-  per  .00001078q-  "•,  or  .001558q- in-  of  section,  which 


FRICTION.  307 

enables  us  to  determine  their  breadth  according  to  the 
thickness. 


254.  Friction  of  Journals. — Besides  the  experiments 
previously  reported,  upon  the  friction  of  plane  surfaces,  I 
have  made  a  great  number  upon  that  of  journals,  by 
means  of  a  rotating  dynamometer  with  a  plate  and  style, 
the  first  apparatus  of  the  kind,  but  which  it  is  not  worth 
while  to  describe  here. 

The  axle  of  this  dynamometric  apparatus  was  hollow 
and  of  cast-iron.  It  could  receive,  by  means  of  holders 
exactly  adjusted,  a  change  of  journals  of  different  mate- 
rials and  diameters.  Its  load  was  composed  of  solid  cast- 
iron  discs  weighing  331lbs-  each,  whose  number  could  be 
increased  so  as  to  attain  a  load  of  more  than  3042lbs-  A 
pulley,  the  friction  of  whose  axle  was  slight,  and  which 
transmitted  the  motion  by  the  intervention  of  a  spring, 
received  by  a  belt,  the  motion  of  a  hydraulic  wheel,  and 
the  difference  of  tension  of  the  two  parts  of  the  belt  was 
measured  by  the  dynamometer  with  the  style. 

We  used  journals  from  .11  to  .22ft-  in  diameter.  The 
velocities  varied  in  the  ratio  of  1  to  4.  The  pressures 
reached  4145lb%  and  within  these  extended  limits  we 
have  proved  that  the  friction  of  journals  is  subject  to  the 
same  laws  as  that  of  plane  surfaces.  But  it  is  proper  to 
observe,  that  from  the  form  itself  of  the  rubbing  body, 
the  pressure  is  exerted  upon  a  less  extent  of  surface,  ac- 
cording to  the  smallness  of  the  diameter  of  the  journal, 
and  that  unguents  are  more  easily  expelled  with  small 
than  with  large  journals. 

This  circumstance  has  a  great  influence  upon  the 
intensity  of  friction,  and  upon  the  value  of  its  ratio  to  the 
pressure.  The  motion  of  rotation  tends,  of  itself,  to  expel 
certain  unguents,  and  to  bring  the  surfaces  to  a  simply 
unctuous  state.  The  old  mode  of  greasing,  still  used  in 
many  cases,  consisted  simply  in  turning  on  the  oil,  or 


308  FEICTION. 

spreading  the  lard  or  tallow  upon  the  surface  of  the  rub- 
bing body,  and  in  renewing  the  operation  several  times 
in  a  day. 

We  may  thus,  with  care,  prevent  the  rapid  wear  of 
journals  and  their  boxes  ;  but,  with  an  imperfect  renewal 
of  the  unguent,  the  friction  may  attain  .07,  .08,  or  even  .1 
erf  the  pressure. 

If,  on  the  other  hand,  we  use  contrivances  which 
renew  the  unguents  without  cessation,  in  sufficient  quan- 
tities, the  rubbing  surfaces  are  maintained  in.  a  perfect 
and  constant  state  of  lubrication,  and  the  friction  falls  as 
low  as  .05  or  .03  of  the  pressure,  and  probably  still  lower. 
The  polished  surfaces  operated  in  these  favorable  condi- 
tions, became  more  and  more  perfect,  and  it  is  not  sur- 
prising that  the  friction  should  fall  far  below  the  limits 
above  indicated. 

These  reflections  show  how  useful  are  oiling  fixtures 
in  diminishing  the  friction,  which,  in  certain  machines, — 
as  mills  with  complicated  mechanism — consume  a  consid- 
erable part  of  the  motive  work.  "We  cannot,  then,  too 
much  recommend  the  use  of  appliances  to  distribute  the 
unguent  continuously  upon  the  rubbing  surfaces  of  ma- 
chines, and  it  is  not  surprising  that  a  great  number  of 
dispositions  have  been  proposed  for  this  purpose  within  a 
few  years.  We  should  be  careful  to  select  those  which 
only  expend  the  oil  during  the  motion,  excluding  those 
which  feed  by  the  capillary  action  of  a  wick  of  thready 
substances.  These  constantly  drain  the  oil  even  during 
the  repose  of  the  machine,  thus  consuming  it  at  a  pure 
loss. 

255.  Results  of  experiments. — In  the  following  table 
will  be  found  some  of  the  results  of  experiments  in  sup- 
port of  the  preceding  considerations  : 


FRICTION. 


TABLE. 

Experiments  upon  the  friction  of  cast-iron  journals  upon 
cast-iron  "bearings. 


» 

'3 

+s 

J 

§ 

£ 

§ 

0.2 

sj 

& 

o 

1 

£   0 

^>  a 

ll 

s«s- 

"8 

£»«:* 

i!i 

Kemarks. 

1 

« 

0  § 

•^"d 

<o*J  £ 

1 

3 

?  2 

•«DcS 

2     ^ 

I 

2 

^ 

1 

ft. 

ft. 

Ibs. 

0.1961 

r 

0.082 

0.222 

0.082 

In  these  experiments  the 

0.488  V 
0.445 

2269.4^ 

0.082 
0.079 

oil  was  poured  only  upon  the 
surface  of  the  journals. 

0.345  J 

I 

0.079 

0.081 

r 

0.2121 

r 

0.054  = 

In  these  experiments  the 

0.328  J 

0.262  1 
0.409  f 

2269.4^ 

0.052 
0.052 

oil    was   poured    ceaselessly 
upon  the  rubbing  surfaces. 

I 

0.488  J 

I 

0.052 

Mean 

0.053 

0.177) 

s  j 

0.429  ) 
0.409  [ 
0.465  ) 

2241.8  -j 

0.101  ) 
0.109  V 
0.101  ) 

In  these  experiments  the 
oil  was  expelled  hy  the  pres- 
sure, and  the  surfaces   were 

simply  very  unctuous. 

Meau 

0.104 

• 

r 

0.190^ 

- 

0.0701 

0.268 

0.069 

In  these  experiments  the 

0.177- 

1  - 

0.328 
0.393 

- 

2240.7- 

0.075 

0.084  ' 

surfaces  themselves  supplied 
the  lard. 

" 

0.445 

0.070 

0.465  j 

0.060 

: 

; 

0.222^ 

0.049s 

. 

0.331 

0.050 

In  these  experiments  the 

0.328- 

a   - 

0.380 

, 

4157.- 

0.052   - 

unguent  was  renewed. 

H 

0.409 

0.040 

0.429^ 

0.042 

' 

very,  low.   ' 

0.037  = 

0.150 

0.039 

. 

0.238 

0.025 

In  these  experiments  the 

0.328- 

OS     " 

0.321 

- 

2276.  - 

0.026  • 

unguent  was  continually  re- 

H 

0.321 

0.035 

newed. 

0.380 

0.026 

. 

. 

0.492  J 

0.832 

810  FRICTION. 

The  examples  contained  in  this  table  suffice  to  show 
that  the  friction  of  journals  is  in  itself  subject  to  the  same 
laws  as  that  of  plane  surfaces ;  but  they  also  show  the 
great  influence  which  the  constant  renewal  of  the  unguent 
possesses  in  diminishing  the  value  of  the  ratio  of  the  fric- 
tion to  the  pressure,  which  sometimes  falls  as  low  as 
.025. 

We  see  also  that  the  diameter  of  the  journals  seems 
to  have  some  influence  upon  the  more  or  less  complete 
expulsion  of  the  unguent,  and  consequently  upon  the 
friction,  so  that  the  dimensions  to  be  given  them  should 
not  be  determined  from  a  consideration  solely  of  their 
resistance  to  rupture. 

Recapitulating,  the  summary  of  the  experiments  which 
I  have  made  upon  the  friction  of  journals,  shows  that  it 
is  nearly  the  same  for  woods  and  metals  rubbing  upon 
each  other,  and  that  its  ratio  to  the  pressure  may,  accord- 
ing to  the  case,  take  the  values  given  in  the  following 

TABLE. 


STATE  OF  SUEFACES. 


With  rotten-stone 
and 
perfectly  greased 

Continually  supplied 
unguent 

Greased  from  time 
to  time 
/ 

Unctuous 

0.025  to  0.030 

0.050 

0.07  to  0.08 

0.150 

256.  Advantage  of  granulated  metals. — It  is  not  true, 
as  is  generally  supposed,  that  the  friction  is  always  less 
between  substances  of  different  kinds  than  between  those 
of  the  same  kind.  But  it  is  well  generally  to  select  for 
the  rubbing  parts  granulated  rather  than  fibrous  bodies, 
and  especially  not  to  expose  the  latter  to  friction  in  the 
direction  of  the  fibres,  because  the  fibres  are  sometimes 


FRICTION.  311 

raised  and  torn  away  throughout  their  length.  In  this 
respect,  fine  cast-iron,  which  crystallizes  in  round  grains, 
as  well  as  cast-steel,  are  very  suitable  bodies  for  parts 
subjected  to  great  friction.  Thus,  for  several  years  past, 
a  cast-iron  packing  has  come  into  very  general  use  for  the 
pistons  of  steam-engines.  If  for  the  boxes  of  iron  or  cast- 
iron  axles,  brass  continues  in  use,  it  is  chiefly  because  it 
is  less  hard,  and  wears  out  before  the  axles,  and  because 
it  is  easier  to  replace  a  box  than  an  axle. 

257.  Remarks  upon  very  light  mechanisms. — In  very 
light  mechanisms,  and  especially  with  very  rapid  motion, 
the  viscosity  of  the  unguent  may  offer  a  resistance  similar 
to  that  produced  by  friction  proper ;  in  such  cases,  the 
results  of  experiments  made  under  considerable  pressures 
in  relation  to  the  surfaces  of  contact,  should  only  be  ap- 
plied with  extreme  caution. 

258.  Use  of  the  results  of  experiments. — The  results 
obtained  from  the  experiments  at  Metz  are  resumed  in 
the  three  following  tables,  which  give  the  ratio  of  the 
friction  to  the  pressure,  for  all  the  substances  employed 
in  constructions.    The  first  of  these  tables  relates  to  plane 
surfaces  which  have  been  some  time  in  contact.     The 
values  which  it  gives  for  the  ratio/* of  friction  to  the  pres- 
sure, should  be  employed  whenever  we  are  to  determine 
the  effort  necessary  to  produce  the  sliding  of  two  bodies 
which  have  been  some  time  in  contact.     Such  is  the  case 
with  the  working  of  gates  and  their  fixtures,  which  are 
used  only  at  intervals  more  or  less  distant. 


312 


FRICTION. 


TABLE   No.  I. 

Friction  of  plane  surfaces  which  have  been  some  time  in 

contact. 


Kind 
of 
surfaces  in  contact. 

Disposition 
of 
the  fibres. 

Condition 
of 
the  surfaces. 

Ratio 
of  friction 
to  pressure  f. 

parallel               .   . 

without  unguent  .  .  . 

062 

do 

rubbed  with  dry  soap 

0.44 
054 

Oak  on  elm  

do 
wood    upright    on 
wood  flatwise  
parallel  

moisten'd  with  water 

without  unguent  
do 

0.71 

0.43 
0.88 

Elm  on  oak                                •< 

do 
do 

do 
rubbed  with  dry  soap 

0.69 
0.41 

Ash,  pine,  beech,  sorb  on  oak 

perpendicular 
parallel              .... 

without  unguent  — 
do 

0.57 
0.53 

Tanned  leather  on  oak  < 
Black  curried  (  on  plane  oak  sur- 

the  leather  flatwise 
the  leather  on  edge 

parallel            .  ... 

do 
do 
moisten'd  with  water 
without  unguent  .... 

0.61 
0.43 
0.79 
0.74 

leather      •<     face, 
or  belt       (  on  oak  drum. 

perpendicular  

do 

0.47 

Hemp  matting  on  oak                 < 

parallel  

without  unguent  — 

0.50 

do 
parallel 

moisten'd  with  water 
without  unguent. 

0.87 
080 

parallel  

do 

0.62 

Oast-iron  on  oak. 

do 
parallel 

moistn'd  with  water 
do 

0.65 
065 

Brass  on  oak  

parallel  

without  unguent.  .  .  . 

0.62 

Oxhide  for  piston  packing  on 

flatwise      

moisten'd  with  water 

0.62 

cast-iron 

on  edge               .  . 

with  oil  lard,  tallow 

0.12 

Black  curried  leather,  or  belt 

flatwise  

$  without  unguent  . 

0.28 

upon  cast-iron  pulley  
Cast-iron  upon  cast-iron. 

do 

(  moist'd  with  water 
without  unguent.  .  .  . 

0.38 
0.16* 

Iron  upon  cast-iron  

do 

do 

0.19 

Oak,  elm,  yoke-elm,  iron,  cast-  ) 
iron   and.  brass   sliding  two  > 

flatwise  

spread  Avith  tallow, 

O.lOt 

and  two  one  upon  the  other.  \ 
Calcareous  oolite  upon  oolite  ) 
limestone                              .  ) 

do 

with  oil,  or  lard.  .  . 
without  unguent.  .  .  . 

0.15J 
0.74 

Hard  calcareous   stone    called  J 
Muschelkalk    upon   oolite     > 
limestone                  .            .  .  ) 

do 

do 

0.75 

Brick  on  calcareous  oolite  

do 

do 
do 

0.67 
063 

Iron  on           do            do  
Hard  muschelkalk  on  muschelk. 
Calcareous  oolite  upon       do.  ... 
Brick  on                              do  
Iron  upon                           do  
Oak  on                                do  — 

Calcareous  oolite  on  calcareous  1 
oolite                      .                 [ 

do 
do 
do 
do 
do 
do 

do 

do 
do 
do 
do 
do 
do 
("with  mortar  three 
J  parts  fine  sand  and 
1  1  part  of  hydraulic 

0.49 
0.70 
0.75 
0.67 
0.42 
0.64 

1     0.74§ 

1  lime. 

*  The  surfaces  being  somewhat  unctuous. 

t  When  the  contact  had  not  been  long  enough  to  press  out  the  unguent. 
$  When  the  contact  had  been  long  enough  to  press  out  the  unguent  and  bring  the 
surfaces  to  an  unctuous  state. 

§  After  a  contact  of  from  10  to  15  minutes. 


313 


TABLE    No.    II. 

Friction  of  plane  surfaces  in  motion  upon  each  other. 


Surfaces  in  contact. 

Position  of  fibres. 

State  of  surfaces. 

Ratio  of 
friction  to 
pressure  f. 

Oak  on  oak  < 

parallel 

'I  without  unguent.  .  . 
rubb'd  with  dry  soap 
}  without  unguent  
wet  with  water  
J  without  unguent  
do 
do 
do 
do 
do 
wet  with  water  

0.48 
0.16 
0.34 
0.25 
0.19 
0.43 
0.45 
0.25 
0.36  to  0.40 
0.62 
0.26 
0.21 
0.49 
0.22 
0.19 
0.62 
0.25 
0.20 
0.27 
0.30  to  0.35 
0.29 
0.56 
0.86 

0.23 
0.15 
0.52 
0.33 
0.38 
0.44 

0.18t 
0.15t 
0.31 
0.20 
0.22 
0.16$ 

07to08§ 

0.15 
0.64 
0.67 
0.65 
0.38 
0.69 
0.38 
0.65 
0.60 
0.38 
0.24 
0.80 

do 
perpendicular  
do 
upright  on  flatwise, 
parallel  ) 

Elm  on  oak    .        < 

perpendicular  > 
parallel  3 

Ash,  pine,  beach,  wild  pear  and 
goto  on  oak    

,0 

do           1 
do           j 

do 
do 
do 
do 
flatwise  on  edge..  < 
I 
flatwise    and   on  J 
edge  | 

rubbed  with  dry  soap, 
without  unguent.  
wet  with  water  

rubbed  with  dry  soap, 
without  unguent  
do 
do 
do 
do 
wet  with  water 

Iron  on  elm    .          

Black  curried  leather  on  oak  
Tanned  leather  on  oak  $ 

Tanned  leather  upon  cast-ironl 
and  brass  i 

without  unguent  
wet  with  water  

unctuous  and  wet  with 

Hemp   strips    or    cords   upon  $ 
oak                                            ; 

[ 

spread  with  oil  

without  unguent.  
wet  with  water 

perpendicular  
parallel 

Wild  pear  on  cast-iron  

do 
do 
do 
do 
do 

do 

do 

do 

do 
do 
wood  upright  
parallel  

do 
do 
do 
do 
wet  with  water  

Iron  upon  iron  

Iron  upon  cast-iron  and  brass  
Cast-iron  on  cast-iron  and  brass. 
Cast-iron  on  cast-iron  

(  OH  brass  

Brass  <  on  cast-iron  .  . 

do 
do 

lubricated  in  the  usual 
way  with  tallow,lard, 
soft  coom,  &c  

r  on  iron 

Oak,  elm,  yoke-elm,  wild  pear,  f 
cast-iron,  iron,  steel,  steel  and  j 
brass,  sliding  upon  each  other  j 

Calcareous  oolite  on  calc.  oolite  < 

Muschelkalk  upon      do        do.. 
Common  brick  upon  do        do  .  . 
Oak  on  oolitic  limestone  
Forg'd  iron  upon  oolitic  limestone 
Muschelkalk  upon  muschelkalk.  . 
Oolitic  limestone  upon       do.  ... 
Common  brick  on               do  .... 
Oak  on                                do  
Iron  on                               do  .  .  \ 

C  slightly  unctuous  to 
'     the  touch       .  ... 

(  without  unguent.  .  .  . 
do 
do 
do 
do 
do 
do 
do 
do 
j               do 
(wet  with  water  

wood  upright  
parallel  

do 

*  Surfaces  worn  when  there  was  no  unguent, 
t  The  surfaces  still  being  slightly  unctuous. 
%  The  surfaces  slightly  unctuous. 

§  When  the  unguent  is  constantly  supplied,  and  uniformly  laid  on,  this  ratio  may  be 
lowered  to  0.05. 


314: 


FRICTION. 


TABLE    No.   III. 

Friction  of  journals  in  motion  upon  their  pillows. 


Surfaces  in  contact. 

State  of  surfaces. 

Ratio  of  friction  to  the  pressure 
when  the  unguent  is  renewed. 

in  the  common 
way. 

continuously. 

Cast-iron  journals  in  cast- 
iron  bearings  

'  unguents  of  olive  oil,  of  lard,  of 
tallow,  or  of  soft  coom  

0.07  to  0.08 

0.08 
0.054 
0.14 
0.14 

o.or  to  o.os 

0.16 
0.16 
0.19 
0.18 

0.10 
0.14 

O.OT  to  0.08 
O.OT  to  0.08 
0.09 
0.19 
0.25 
0.11 
0.19 
0.10 
0.09 

0.12 
0.15 

0.030  to  0.054 

0.03  to  0.054 

* 
t 
0.090 

0.030  to  0.054 
0.030  to  0.054 

* 

0.030  to  0.052 
0.07 

with  the    same  unguents    and 
moistened  with  water  

Cast-iron  journals  on  brass 

unctuous 

unctuous  and  wet  with  water.  . 
"unguents  of  olive  oil,  of  lard,  of 
tallow,  and  of  soft  coom  

\  unctuous  

Cast-iron  journals  on  lig- 
num vitae  bearings  

Wrought-iron  journals  on 
cast-iron  bearings  

unctuous  and  wet  with  water.  .  . 
slightly  unctuous 

C  without  unguent 

unguents  of  oil  or  lard  

<  unctuous  with  oil  or  lard 

unctuous,  with  a  mixture  of  lard 
l_     and  black  lead 

j  unguents  of  olive  oil,  tallow,  lard, 
\     or  soft  coom  

Wrougbt-iron  journals  on 
brass  bearings  

["unguents  of  olive  oil,  tallow,  lard 
j  unguents  of  soft  coom  

|  unctuous,  and  wet  with  water.  . 
(^slightly  unctuous 

Iron  journals   on   lignum 
vitse  bearings  

J  unguents  of  oil  or  lard  

1  unctuous  

Brass    journals   on    brass 
bearings 

J  unguents  of  oil  

unguent  of  lard              .... 

Brass  journals  on  cast-iron 
cushions  
Lignum  vitse  journals  on 
cast-iron  cushions.  
Lignum  vitae  journals   on 
lignum  vitse  cushions  .  .  . 

unguents  of  oil  or  tallow.  .   .   . 

unguents  of  lard  

unguent  of  lard  

*  The  surfaces  began  to  wear. 

t  The  wood  being  slightly  unctuous. 

$  The  surfaces  began  to  wear  away. 


Table  No.  2  relates  to  plane  surfaces  in  motion  upon 
each  other,  table  No.  3  applies  to  journals  in  motion  upon 
their  bearings.  The  values  given  by  these  tables  ought 
not  to  be  used  except  to  calculate  the  friction  of  two  sur- 
faces in  motion  upon  each  other,  after  the  period  in  which 
the  coefficient  of  friction  at  the  starting  has  been  introduced. 


FRICTION. 


315 


259.  Application  to  gates.  —  Let  L  be  the  horizontal 
width  of  a  gate  under  a  certain  head  of  water,  and  H'  the 
head  or  height  of  level  above  a  horizontal  section  of  this 
gate,  of  a  thickness  A'  infinitely  small.  The  pressed  sur- 
face of  this  element  will  be  LA',  and  the  pressure  which 
it  will  experience  will  be  62.32LH/A'.  The  total  pres- 
sure upon  the  entire  surface  of  the  gate  being  equal  to  the 
sum  of  all  the  similar  pressures  upon  each  of  the  elements, 
will  have  for  its  value 

62.32L(H/A/+H//A//+H///A///+&c.). 

Now,  the  products  LH'A',  LH"A",  etc.,  are  the  mo- 
ments of  the  elementary  surfaces  LA7,  LA",  etc.,  in  rela- 
tion to  the  plane  of  the  level,  and  their  sum  is  equal  to 
the  moment  of  the  whole  surface  equal  to  LEU.  Calling 
E  the  height  of  the  gate  pressed,  and  H  the  distance  of 
the  centre  of  gravity  from  the  surface  of  the  level,  or  the 
head  upon  the  centre  of  the  figure.  Then  the  total  pres- 

sure is 

62.32  LEH, 

and  the  friction  which  results  against  the  slides  of  this 

gate  is 

62.32/.  LEH, 

/being  the  ratio  of  the  friction  to  the  pressure  for  the 

surfaces  in  contact,  a  ratio  whose 

value  should  be  taken  from  the 

first  table,  if  we  are  to  calculate 

the  effort  required  to  put  the 

gate  in  motion. 


fl 


EXAMPLE.-]?  L=6.56ft, 
E=1.148ft-,  H=4.92ft-,  the  first 
table  gives  for  a  wood  gate  of 
oak  sliding  with  crossed  fibres 
upon  oak  wet  with  water  y=0.71  ;  we  have  then  for  the 
friction  62.32  x  0.71  x  6.56  x  1.148  x  4.92ft-=1639.4:lb9- 


FHJ.IOS. 


316  FKICTION. 

The  effort  should  be  transmitted  in  the  direction  of 
the  racks  fixed  upon  the  gate ;  and  as  it  is  considerable,  it 
will  be  proper  to  arrange  a  kind  of  screw-jack,  suitably 
proportioned,  for  the  establishment  of  which  we  may  take 
as  the  effort  to  be  exerted  by  a  man  upon  the  winch,  at 
any  instant,  from  55  to  66  pounds  at  most,  and  during  the 
motion  from  22  to  26.5  pounds. 

When  the  gate  is  in  motion,  the  effort  to  be  trans- 
mitted to  the  racks  is  much  less,  because  the  ratio  of  the 
friction  to  the  pressure  diminishes,  and  is  reduced  for  a 
gate  with  moistened  slides  to  0.25,  which  gives  for  the 
friction  during  motion 

62.32  x  0.25  LEH=62.32  x  0.25  x  6.56  x  1 .148  x  4.92 

=577.2lbs- 

at  the  first  instant,  and  a  value  decreasing  with  the  raising 
of  the  gate,  or  as  the  head  H  upon  its  centre  is  lessened. 

We  hardly  need  to  say  that,  in  working  the  gate  we 
must  calculate  for  the  maximum  effort. 

260.  Application  to  saw  frames. — If  we  have,  for 
example,  the  frame  of  a  saw  for  veneering,  subjected  to  a 
pressure  of  110.274lb%  and  provided  with  iron  strips 
sliding  in  brass  grooves,  greased  with  lard,  we  have,  if 
the  surfaces  are  well  lubricated,  for  the  friction, 

0.07xll0.274=7.719lb% 
and,  if  they  are  unctuous, 

0.15xll0.274=16.54lbs- 

If  the  stroke  of  the  frame  is  3.936ft>,  and  the  number 
of  strokes  180  in  1',  the  space  described  in  V  will  be 
11.81ft',  and  the  work  consumed  by  the  friction  of  the 
frame  in  V  will  be,  in  the  first  case, 

2  x  11.81  x  7.719=182.32lbs-ft-=4  horse  power  nearly, 

o 


FRICTION.  317 

in  the  second  case 

2  x  11.81  x  16.54r=390.66lbs-  ft-=|  horse  power  nearly. 

3 

261.  Application  to  journals. — To  calculate  the  work 
consumed  by  the  friction  of  the  journals  of  a  revolving 
axle,  we  begin  by  seeking  the  resultant  of  the  forces  act- 
ing around  this  axle,  and  decompose  this  into  two,  the 
one  horizontal  and  the  other  vertical,  and  we  take  sepa- 
rately the  resultant  of  each  of  these  groups.  Calling  X 
the  sum  of  the  horizontal  components,  Y  the  sum  of  the 
vertical  components,  the  general  resultant  will  be 


and  the  friction  produced  by  it  will  be 


The  theorem  of  M.  Poncelet,  already  cited  in  "No.  227, 
informs  us  that  when  we  do  not  know  the  order  of  mag- 
nitude of  X  and  Y,  we  may  calculate  to  nearly  -  of  the 

value  of  the  radical  by  the  formula  0.83  (X+Y),  and  that 
if  we  know  beforehand  that  one  of  the  terms,  X  for  exam- 
ple, is  greater  than  the  other,  which  is  most  usually  the 

case,  we  shall  have  the  value  of  the  radical  to  —   nearly, 

2o 

by  the  expression  0.96  X+0.4Y. 

Suppose,  for  example,  that  we  have  a  hydraulic  bucket- 
wheel  weighing  88219  pounds,  transmitting  a  useful  effect 
of  50  horses'  power  to  the  exterior  circumference,  and 
imparting  motion  to  a  pinion,  so  that  the  useful  resistance 
may  be  horizontal  and  represented  by  Q.  Suppose  the 
radius  of  the  wheel  R=:9.84:ft-,  the  velocity  at  its  circum- 
ference to  be  5.249ft>,  and  the  radius  of  the  gearing  wheel 


318  FEICTION. 

K'=6.56ft>    The  effort  P  transmitted  to  the  circumference 
of  the  wheel  will  be 


The  pressure  upon  the  journals  of  the  hydraulic  wheel 
will  be 


or,  since  M=88219lb%  and  consequently  M+P  is  greater 
than  Q,  we  may  take  for  an  approximate  value  of  the 

radical  to  —  nearly 
25 

0.96  (M+P)+0.4Q. 

For  uniform  motion,  the  moment  of  the  power  P  must 
be  equal  to  the  sum  of  the  moments  of  resistances.  We 
have  then,  in  calling  r  the  radius  of  the  journal  =0.393ft>, 
/=0.07, 

5239lbs-  x  9.84=  Q  x  6.56+0.96  (0.07)  (88219+5239)  (0.393) 

+0.4  (0.07)  (Qx  0.393); 
whence 

n  _5239  x  9.84-0.96  x  0.07  x  93458  x  0.393__AQ  Qlbs 
***  6.56x0.4x0.07x0.393 

while  if  we  had  neglected  the  friction  of  the  journals,  we 
should  have  found 


The  velocity  of  the  gearing  wheel  being 

5.249  x?=3.499ft- 
o 


FRICTION.  319 

The  work  transmitted  to  this  circumference  in  V  is 

7469.8lbs-  x  3.499=:26137lbs-ft-=47.5  horses'  power. 
The  loss  by  the  friction  of  the  journals  is  then 
50.00  horses'  power  -47.5=2.5  H.  P. 

If  the  surfaces  of  the  journals  had  not  been  unctuous 
the  loss  would  have  been  double. 

The  space  described  by  the  rubbing  points,  being  one 
of  the  factors  of  work  consumed  by  the  passive  resistance, 
it  is  important  to  diminish  it  the  most  possible,  and  con- 
sequently to  give  the  journals  only  such  dimensions  as 
will  ensure  a  proper  strength. 

To  calculate  their  diameter  in  the  establishment  of 
the  wheel,  we  disregard  the  friction,  which  will  give  us  a 
first  value  of  Q=785Slbs-,  a  little  too  much,  and  conse- 
quently for  the  resultant  of  the  efforts  to  which  the  jour- 
nal is  subjected 


<Y/(93458)2-K7858.6)2=93787lbs- 

Each  journal  supports  then  nearly  46893lb8>  of  pres- 
sure, and  its  diameter,  calculated  by  the  formula  for 
journals  of  hydraulic  wheels,  will  be 

*  d=. 00364 -v/46893; 
whence 


This  is  the  value  which  we  have  adopted  in  the  pre- 
ceding calculation. 

*  In  original  / — p — 

~  r    368156* 
In  English  measures 


(=/ 


P  x  3.2809' 


320  FRICTION. 

262.  Axles  of  wagons — We  should  calculate  in  a 
similar  manner  the  friction  of  the  axles  of  wagons  against 
their  boxes ;  observing  that  it  is  the  box  which  slides 
around  the  axle,  and  that  the  path  described  by  the 
rubbing  points  is  at  the  circumference  of  the  box,  and  its 
arm  of  lever  the  radius  of  the  box. 


KIGIDITY  OF  COEDS. 


263.  Rigidity  of  cords. — When  a  cord  solicited  at  its 
extremity  by  a  weight  or  an  effort  of  tension,  is  passed 
over  a  cylinder  or  a  pulley,  movable 

around  its  axis,  it  experiences  a  diffi- 
culty in  bending  owing  to  its  stiffness, 
and  the  curve  it  takes  is  of  a  greater 
radius  than  that  of  the  cylinder,  so 
that  the  direction  of  the  part  prolonged 
passes  at  a  greater  distance  from  the 
axis,  than  the  radius  of  the  wheel  in- 
creased by  that  of  the  cord.  From 
this  it  follows  that  the  moment  of  ten- 
sion of  this  part  is  increased  by  a  cer- 
tain quantity  arising  from  the  resist- 
ance of  the  cord  to  flexure. 

This  resistance,  known  by  the  name  of  rigidity  of 
cords,  has  been  experimentally  investigated  by  Amon- 
tons,  and  more  lately  by  Coulomb,  who  made  use  of  the 
apparatus  contrived  by  his  predecessor,  and  of  another 
arrangement  similar  to  that  described  in  No.  223,  which 
served  for  his  experiments  upon  rolling.  We  may  gather 
sufficiently  exact  ideas  of  these  researches,  by  confining 
ourselves  solely  to  those  of  Coulomb,  which,  though  in- 
complete, are  the  best  we  have  upon  the  subject. 

264.  Experiments   of  Coulomb,  with  the  apparatus 
of  Amontons. — In  this  apparatus,  a  free  roller  LM  is  en- 

21 


FIG.  104, 


322 


RIGIDITY    OF   COEDS. 


circled  by  one  turn  of  each  of  the  two  portions  of  a  cord, 
which  passes  over  two  pulleys  A  and  B  made  fast  to  a 
beam.  At  the  ends  C  and  D  of  this  cord  are  two  hooks 
sustaining  a  platform  loaded  with  a  weight.  The  roller  is 


Q 

FIG.  105. 


a 

FIG.  106. 


placed  horizontally,  and  the  turn  of  each  of  the  portions 
encircling  it,  is  arranged  symmetrically  in  respect  to  each 
other.  Midway  between  these  turns,  a  flexible  thread 
passes  round  the  roller,  to  which  it  is  fixed  at  one  end, 
while  it  supports  at  the  other  a  small  plate,  in  which  is 
placed  a  weight  q,  necessary  for  the  slow  descent  of  the 
roller. 

In  this  movement,  the  lower  portion  of  the  cord  is 
enrolled  upon  the  roller,  and  the  upper  part  is  unrolled. 

The  tension  of  each  part  is  equal  to  the  half  -~  of  the  load 

2 

of  the  platform.  Moreover,  it  is  readily  seen,  that  the 
space  described  by  the  motive  weight  q  will  be  double 
the  space  described  by  the  enrolled  parts  of  the  cord.  In 
fact  when  the  roller  has  descended  from  the  point  of  con- 
tact a  to  the  point  5,  (Fig.  107),  it  is  evident  that  the  arc 
of  the  enrolled  cord,  or  the  space  described  in  the  direc- 
tion of  the  resistance  to  rolling  will  be  ab.  It  is  clear 


RIGIDITY   OF   COEDS. 


that  in  this  displacement,  the  point  of  the  thread  of  sus- 
pension of  the  motive  weight  <£,  which  shall  have  come 
into  the  vertical  or  in  contact,  will  be  a 
point  d  placed  at  a  distance  cd  equal  to  the 
arc  c^'=arc  db'=db)  which  the  roller  itself 
has  described.  Then  the  weight  will  have 
descended  db  by  the  translation  of  the  roller, 
and  cd'=ab  by  its  rolling,  or  %db  in  all. 

The  work  developed  by  this  weight  will 
be  qx2al>=q.  Da,,  calling  D  the  mean 
diameter  of  the  roller,  and  al  the  angle  de- 
scribed at  the  unit  of  distance,  while  the 
work  developed  by  the  rigidity  Bj  of  each 
portion  will  be  equal  to  the  rigidity  itself 
multiplied  by  the  space  described,  or  to 


D 


FIG.  107. 


We  shall  have  then  at  the  moment  of 
equilibrium,  or  when  the  motion  is  very  slow  and  nearly 
uniform,  by  reason  of  the  resistance  of  the  two  portions 
of  the  cord, 

qD=^1-  ;  whence  q=E>^ 
2i 

that  is  to  say,  the  motive  weight  is  equal  to  the  resistance 
which  each  of  the  two  parts  oppose  to  the  enrolling. 

265.  Results  of  the  experiments  of  Coulomb. — Before 
taking  other  steps,  we  introduce  in  the  following  table  the 
data  and  results  of  some  experiments  made  by  Coulomb, 
with  the  apparatus  of  Amontons,  limiting  ourselves  to  the 
transformation  of  the  old  measures  into  the  new. 

The  cords  of  6  threads,  of  15  and  30  threads,  which  he 
used,  were  rolled  on  rollers  1.09in%  2.18,  and  4.37ins-  diam- 
eter, and  the  total  tensions  varied  from  55  to  2205  pounds, 
the  motor  weights  varying  also  through  extended  limits. 


324 


RIGIDITY   OF   CORDS. 

TABLE. 


Results  of  experiments  of  Coulomb  upon  the  rigidity  of 
cords,  made  with  the  apparatus  of  Amontons. 


03   O 

9"te 
*•  a 

2 

VALUES  OF  THE  MOTOR  WEIGHTS  FOUND   FOR  CORDS   OF  THE  DIAMETERS 

d=0.02SSft.  or  6  threads 
rolled  round  rollers  with 
diameters  D  equal  to 

<fc=0.0472ft.  or  15   strands 
wound  round  rollers  with 
diameters  D  equal  to 

rf:=0.0656ft.  or  30  strands 
•wound  on  rollers  with 
diameters  D  equal  to 

0.088ft. 

0.177ft. 

0.354ft. 

0.088ft. 

0.177ft. 

0.854ft. 

0.177ft. 

0.854ft. 

0.531ft. 

Ibs. 
26.98 
134.95 
242.91 
458.93 
674.74 
1106.58 

Ibs. 
2.159 
11.877 
18.354 
33.468 
46.423 

Ibs. 

4.318 
7.108 
12.955 
33.468 
46.423 

Ibs. 

6.153 
9.717 

11.877 

Ibs. 
7.558 
23.751 
81.064 
70.174 
99.324 

Ibs. 
3.454 
9.717 
18.354 
33.468 
44264 

Ibs. 
1.835 
5.897 
7.558 
14.036 
20.235 
29.149 

Ibs. 
11.877 
22.725 
31.309 
50.741 
72.333 

Ibs. 
5.399 
9.177 
15.114 
24.832 
33.468 
53.979 

Ibs. 
36.706 

Coulomb  concluded  from  his  experiments,  that  for  the 
same  cord,  the  resistance  to  rolling  varied  in  the  inverse 
ratio  of  the  diameter  of  the  roller,  whence  it  follows  that 
the  product  of  the  resistance,  or  of  the  motor  weight  pro- 
ducing equilibrium  into  the  diameter  of  the  roller,  should 
be  a  constant  quantity,  whatever  may  be  the  magnitude 
of  the  roller.  Let  us  see  if  this  consequence  is  sufficiently 
substantiated,  and  for  this  purpose  we  obtain  the  product 
of  the  diameters  of  the  rollers  and  the  values  of  the  motor 
weights  q,  for  each  cord  and  each  tension  ;  we  thus  have 
the  results  given  in  the  following  table,  which,  in  case  of 
Coulomb's  inferences  being  admitted,  will  represent  the 
resistances  to  rolling  upon  a  cylinder  1  foot  in  diameter. 


VALUES   OF   THE   PRODUCT    qTf   FOR   CORDS   WITH   DIAMETERS   OF 


f  lid 

If 

<fc=0.028  or  6  strands  wo'nd 
on  rollers  with  diame- 
ters D  equal  to 

<Z=0.047ft.  or  15   strands 
wound  on  rollers  with 
diameters  D  equal  to 

^=0.065ft.  or  30   strands 
wound  on  rollers  with 
diameters  D  equal  to 

0.088ft. 

0.177ft. 

0.854ft. 

0.088ft. 

0.177ft. 

0.354ft. 

0.177ft. 

0.354ft. 

0.531ft. 

Ibs. 
26.99 
134.95 
242.91 
458.83 
674.74 
1106.58 

0.1913 
1.0528 
1.6261 
2.9652 
4.1130 

0.7643 
1.2422 
2.2930 
5.9238 
8.2169 

2.180 
8.448 

4.208 

0.6696 
2.1043 
2.7523 
6.2174 

8.8001 

0.6114 
1.7199 
3.2487 
5.9238 

7.8420 

0.6501 
1.9122 
2.6778 
4.9730 
7.1693 
10.3275 

2.1022 
4.0223 
5.5430 
8.9898 
12.8152 

1.9129 
3.2514 
5.3549 
8.7980 
11.8577J 
19.1248 

19.5092 

RIGIDITY   OF   CORDS.  325 

An  examination  of  this  table  shows  that,  for  the  cord 
of  0.065ft-  diameter,  the  values  of  the  product  qD  are 
nearly  equal  for  all  the  diameters  of  the  rollers  used  ;  that 
it  is  nearly  the  same  for  the  majority  of  the  results  fur- 
nished by  the  cord  of  0.(M7ft- ;  but  for  the  cord  of  0.0288ft- 
they  are  very  much  less  in  accordance  with  the  law 
admitted  by  Coulomb. 

Nevertheless,  as  is  usually  done,  in  the  applications  to 
cords  of  diameters  exceeding  0.029ft ,  we  will  admit,  with 
this  philosopher,  till  more  ample  information  is  derived, 
that  the  resistance  to  rolling  varies  in  the  inverse  ratio  of 
the  diameter  of  the  cylinder. 

266.  General  expression  of  the  resistance  to  rolling.— 
Coulomb  has  drawn  from  his  experiments  the  conclusion 
that  the  resistance  to  rolling  may  be  represented  by  two 
terms,  the  one  constant  for  each  cord  and  each  roller, 
which  we  designate  by  the  letter  A,  and  which  he  termed 
the  natural  rigidity,  since  it  depends  upon  the  mode  of 
fabrication  of  the  cord,  and  upon  the  degree  of  the  tension 
of  its  threads  and  strands ;  the  other,  proportional  to  the 
tension  T  of  the  enrolling  strip,  which  is  expressed  by  the 
product  BT,  in  which  B  is  also  a  constant  number  for 
each  cord  and  each  roller. 

In  the  case  of  the  apparatus  of  Amontons  we  have 

T=^  and  the  formula  of  Coulomb  gives 
2=K,=A+B|. 

Thus,  when  the  values  of  the  motor  weight  q,  answer- 
ing to  each  of  the  values  Q  of  the  whole  weight  of  the 
platform,  are  given  by  experiment,  if  we  take  the  total 
weight  Q  of  the  platform  for  abscissa,  and  the  values  q 
for  ordinates  of  a  line  traced  in  joining  all  the  points  so 
obtained,  this  will  be  a  straight  line,  whose  position  and 


320  RIGIDITY   OF   COEDS. 

T> 

inclination  will  furnish  the  values  of  A  and  —  for    each 

2 

cord  and  each  roller. 

M.  Navier,  in  a  discussion  of  the  experiments  of  Cou- 
lomb, which  is  published  in  the  second  edition  of  the 
"  Architecture  hydraulique  "  of  Belidor,  has  attributed  to 
the  constants  A  and  B  particular  values  for  the  different 
diameters  of  the  cords,  which  are  respectively  as  follows : 


DIAMETER 
of  cords  d. 

VALUES  OF  COEFFICIENTS* 

A. 

B. 

0.065ft. 
0.047 
0.0288 

1.6097 
0.4596 
0.0767 

0.031949 
0.018104 
0.007808 

But  for  comparing  M.  Javier's  formula  with  the  re- 
sults of  experiments,  we  must  introduce  in  the  formula 

R=A+BT  the  values  of  T=— ,  to  derive  the  weight  #, 

lound  with  the  apparatus  of  Amontons.  It  is  thus,  we 
have  made  this  comparison  in  the  figures  of  108,  taking 
for  a  graphic  representation  of  the  results  of  experiments, 
the  abscissa  equal  to  the  total  load,  and  the  ordinates 
*equal  to  the  values  of  qD,  or  to  the  resistance  of  rolling 
upon  a  cylinder  1  foot  in  diameter.  Then,  to  compare 
these  results  with  the  values  of  the  coefficients  deduced 
by  M.  JSTavier,  we  have  taken  for  the  same  abscissa  ordi- 
nates equal  to  the  values  of  A+  -  Q,  deduced  from  the 

2t 

values  of  A  and  B  as  given  by  him  in  the  preceding  table. 

An  examination  of  Fig.  Ill  shows  that  the  values  of 

A  and  B  adopted  by  M.  Kavier  accord  very  well  with 

the  results  of  observations  for  the  cord  of  0.065f%  the 

*  The  coefficients  of  M.  Navier  have  been  changed  to  suit  the  English 
units  of  Ibs.  ft. ;  and  the  following  tables  are  worked  for  a  drum  of  1  foot 
diameter,  instead  of  1  metre. 


KIGIDITY    OF   COEDS. 


327 


straight  line  traced  according  to  the  values  of  his  coeffi- 
cients, diverging  but  slightly  from  all  the  points,  obtained 

FIG.  108. 
Fig.  I.— Cord  of  0.0288ft.        Fig.  II.— Cord  or  0.0472ft.        Fig.  III.— Cord  of  0.065Gft. 


100       200      300       400       500       600       700      800 
Total  weight  in  Pounds. 

Straight  lines  indicate  M.  Navier's  formula. 


1000        1100 


graphically  by  figures  directly  derived  from  experiments; 
but  for  the  cords  of  0.(M7ft-  and  0.029f%  the  values  adopted 
by  this  engineer  are  too  small,  especially  for  the  number 
B,  the  points  corresponding  to  the  figures  of  the  table  in 
Figs.  I.  and  II.,  relative  to  the  cords  of  .029ft-  and  .047ft<  be- 
ing all  situated  above  the  straight  line  which  represents 
the  formula.  The  figures  of  M.  Javier  seem  to  have 
been  determined  solely  by  means  of  the  last  series  of 
experiments  made  upon  each  cord,  and  with  the  value  q 
obtained  for  the  greatest  load. 


328 


KIGIDITY    OF    COEDS. 


267.  Other  experiments  of  Coulomb. — Coulomb   also 
made  use  of  another  method  of  experimenting  upon  the 

FIG.  109— See  page  852. 

Fig.  IV. — Battery  wagon  with  gun  upon  the  route  from  Montigny  to  Metz,  dry. 
Fig.  V. — Battery  wagon  with  gun  upon  the  pavement  of  Metz. 
Fig.  VI. — Stage  coaches  without  springs  upon  the  pavement  of  Paris. 
Fig.  VII. — Stage  coaches  with  springs  upon  the  pavement  of  Paris. 

0.020; 
0.01811 
0.016 
0.014 1 
0.012 1 
0.01  Ol 
0.008 


3.38ft 


4.92ft. 


6.56ft.          8.20ft. 
Velocities. 


rigidity  of  cords,  and  their  resistance  to  rolling ;  having 
placed  rollers  upon  the  horizontal  bench  previously  used 
in  his  experiments  on  friction,  he  loaded  them  with  equal 
weights,  suspended  upon  the  two  strips  of  the  cord.  A 
gentle  motion  was  produced  by  weights  upon  a  flexible 
cord,  whose  rigidity  was  not  regarded.  Previous  experi- 
ments having  enabled  him  to  appreciate  the  resistance 
due  to  the  rolling  of  the  rollers,  by  subtracting  it,  he  ob- 
tained that  arising  from  the  rigidity  of  the  cords.  The 
results  of  these  experiments  are  given  in  the  following 
table,  which  is  reduced  to  measures  in  feet. 


KIGIDITY   OF  COEDS. 


329 


TABLE. 

Experiments  upon  the  rigidity  of  cords  made  ~by  Coulomb 
with  movable  rollers  upon  a  horizontal  plane. 


Loads  or  tensions 
Q. 

Values  of  the  resistance  to  rolling  for  cords  with  diameters  of 

<f=0.065ft.   wound  round  rollers 
with  diameter  D  equal  to 

<fc=0.0472ft  on 
rollers  of 

0.5315ft. 

efc=0.02SSft.  on 
rollers  of 

0.5315ft. 

1.0663ft. 

0.5315ft. 

Ibs. 
26.997 
107.958 
215.92 
323.875 
539.790 

Ibs. 
3.778 

11.874 
15.547 

Ibs. 
14.252 

Ibs. 
1.2307 
4.8190 
8.8528 

18.890 

Ibs. 
3.5618 

Still  admitting  with  Coulomb,  according  to  the  results 
of  the  preceding  experiments,  that  the  resistances  to 
rolling  vary  in  the  inverse  ratio  of  the  diameter  of  the 
rollers,  we  shall  have  the  resistance  for  a  roller  of  one  foot 
in  diameter,  by  multiplying  respectively  each  of  those 
entered  in  the  table,  by  the  diameter  of  the  corresponding 
roller.  This  has  been  done  in  the  following  table,  in 
which  is  inserted  the  values  of  the  same  resistance,  calcu- 
lated by  means  of  the  values  of  A  and  B  admitted  by 
Navier,  in  order  to  see  if  the  formula  R=A+BQ  really 
represents  the  results  of  experiments. 


Kesistances  to  rolling  upon  a  drum  one  foot  in  diameter  for  cords  with 
diameters  of 


Loads  or  ten- 
sions Q. 

0.06 
Observed 

by 

Coulomb. 

5ft. 
Calculated 

by 

M.  Navier. 

0.0472ft. 

0.0288ft. 

Observed 

by 

Coulomb. 

Calculated 
by 
M.  Navier. 

Observed 

by 

Coulomb. 

Calculated 
by 

M.  Navier. 

Ibs. 
26.99 
107.96 
215.92 
323.87 
539.79 

Ibs. 

4.028 
*7.713 
12.660 
16.577 

Ibs. 

5.058 
*8.508 
11.957 
18.855 

Ibs. 
0.654 
2.561 
4.705 

10.040 

Ibs. 
0.948 
2.414 
4.369 

10.232 

Ibs. 
1.893 

Ibs. 
1.763 

*  No  data  in  first  table ;  so  equivalents  for  Morin's  entries  are  here  given. 


330  EIGIDITY   OF   COEDS. 

We  see  by  this  comparison  that  the  values  of  the 
coefficients  A  and  B  adopted  by  M.  Navier  for  the  exper- 
imental cords,  lead  very  nearly  to  the  same  values  of  the 
resistance  to  rolling  upon  a  drum  of  one  foot  diameter,  as 
those  given  by  the  preceding  experiments,  and  as  they 
conform  also  with  experiments  made  with  the  apparatus 
of  Amontons,  upon  a  cord  of  0.065  ft.  diameter  and  with 
those  of  .047  ft.  and  .029  ft.,  made  with  heavy  loads,  it 
follows  that  we  may  adopt  these  values  of  A  and  B  for 
white  dry  cords  in  good  condition. 

268.  Extension  of  the  results  of  CoulomVs  experiments 
to  those  of  different  diameters. — To  extend  the  results  of 
Coulomb's  experiments  to  cords  of  different  diameters 
from  those  experimented  with,  M.  Navier  has  explicitly 
admitted  what  Coulomb  but  vaguely  indicated  :  that  the 
coefficients  A  were  proportional  to  a  certain  power  of  the 
diameter  depending  upon  the  condition  of  the  use  of  the 
cords  :  but  this  supposition  appears  to  us  neither  just  nor 
admissible,  for  it  would  lead  to  the  consequence,  that  an 
old  cord  1  foot  diameter,  would  have  the  same  rigidity  as 
a  new  cord,  which  is  evidently  false,  and  moreover  a  com- 
parison of  the  values  of  A  and  B  proves  that  the  power  to 
which  we  must  raise  the  diameter  will  not  be  the  same 
for  the  two  terms  of  the  resistance.* 


*  In  fact,  M.  Navier  supposes  that  A=ad^  and  E=bd^  a  and  &  being  constants,  de- 
pending upon  the  state  of  use  of  the  cord,  and  /u.  an  exponent,  which  should  be  the  same 
for  both  expressions,  and  which  varies  from  2  to  1,  according  to  the  use.  Now,  it  is  appa- 
rent that  if  the  cord  has  a  diameter  <fc=lft.  it  would  always  be  equal  to  1,  whatever  the 
degree  of  its  use,  and  that  then  the  resistance  of  an  old  cord  would  be  the  same  as  that 
of  a  new,  which  is  not  admissible. 

But  further,  M.  Navier  having  given  for  cords  of  diameters 

m.  kil. 

eZ=0  .  02      the  values    ac^=0  .  222460,    &^=0.009738, 
d=Q  .  0144    "         "       a<^=0  .  063514,    M^=Q.Q05518, 
d=Q .  0088    "         "       ad^-Q  .  010604,    &^=0.002380, 
we  deduce  for  the  values 

of  adP  as  a  mean  /u=3.7526  and  rt=531286kil.. 
and  for  those  of  &<#*  as  a  mean  M=1.T174  and  &=8kil.0520. 

It  follows  from  this  that  the  values  of  the  coefficients  A  and  B  cannot  be  of  the  form 
adf-  and  &cZa,  which  Navier  has  assigned  them,  the  exponent  /a  not  being  the  same  fur 
both  quantities,  and  the  two  constant  factors  a  and  &  having  to  vary  with  the  state  of  use 
of  the  cord.  I  have  transcribed  the  French  measures,  as  the  note  serves  no  other  pur- 
pose but  to  show  the  incorrectness  of  Navier's  formula. 


KIGIDITY    OF   COEDS. 


331 


269.  Expression  of  the  rigidity  of  cwds  in  function  of 
the  number  of  strands.  —  Since  then  the  form  proposed  for 
the  expression  of  the  resistance  of  cords  to  rolling,  cannot 
be  admitted,  we  must  seek  another,  and  it  is  natural  to 
try  if  the  factors  A  and  B  may  not  be  expressed  for  white 
cords,  simply  according  to  the  number  of  their  strands,  in 
the  same  way  as  Coulomb  has  done  for  tarred  cords. 

Now,  dividing  the  values  of  A  obtained  by  Xavier, 
for  each  cord,  by  the  number  of  strands  we  find  for 


7i=30    d=0. 


A=1.6097     ^"=0.053657 


n 

A=0.45958  ^=0.030639 
n 

n=  6     d=0.029      A=0.07673  -=0.012788 

n 

"We  see  by  this  that  the  number  A  is  not  simply  pro- 
portional to  the  number  of  strands. 

Moreover,  in  comparing  the  values  of  the  ratio  —  cor- 
responding to  three  cords,  we  find  the  following  results  : 

TABLE. 


DIFFERENCES 

VALUES 

DIFFERENCES 

of  values  of 

NUMBER 

of 

DIFFERENCES 

of  values  of 

J^ 

of 

A 

of  numbers  of  threads. 

A 

ft. 

threads. 

n 

n              by  difference  of 

threads. 

30 

0.053657 

From  30  to  15,  15  threads 

0.023018 

0.001534 

15 

0.030639 

From  15  to  6,  9       do 

0.017851 

0.001983 

6 

0.012788 

From  30  to  6,  24     do 

0.040869 

0.001702 

Mean  difference  per  thread 0.001739 


It  follows  from  this  that  we  may  represent  with  suffi- 
cient accuracy  for  practice  the  values  of  A  given  by 
experiment,  by  the  formula 

A=n-  [0.012788+0.001739  (n  -  6)] =n  [0.002354 
+0.00173971], 


332  RIGIDITY   OF  CORDS. 

an  expression  relating  solely  to  white  and  new  cords,  such 
as  those  with  which  Coulomb  operated. 

As  to  the  values  of  B,  it  seems  to  be  proportional  to 
the  number  of  threads,  for  we  find  for 

B=0.031949     ?=0.001064 
n 

B=0.018104     ?=0.001206 

n 

n=  6     fcO.0288      B=0.007808    ?=0.001301 

n 

Mean    .     .     0.001190 
whence  B=0.001190?z. 

Consequently,  we  may  represent  with  sufficient  exact- 
ness for  practice,  the  results  of  Coulomb's  experiments, 
upon  white  new  and  dry  cords,  by  the  formula 

E=^  [0.002354+0.00173971+0.001190 .  Q]lb% 

which  gives  the  resistance  to  rolling  upon  a  drum  of  one 
foot  diameter,  or  by  the  formula 

K=g  [0.002354  +  0.001739^+0.001190 .  Q]lbs- 

for  a  drum  of  diameter  D. 

270.  Remarks  upon  cords  that  have  been  used. — As 
for  cords  which  have  been  used,  the  rule  given  by  M. 
Navier  cannot  be  admitted,  as  I  have  shown  in  the  pre- 
ceding note,  since  it  gives  for  the  rigidity  of  a  cord  equal 
to  unity,  the  same  as  for  a  new  one  ;  and  it  is  from  hav- 
ing adopted  (in  common  with  other  authors)  this  rule, 
without  discussing  its  elements,  that  I  was  led  to  this 
inadmissible  result,  in  calculating  the  table  of  rigidity  of 
cords  inserted  in  the  third  edition  of  my  "  Aide-Memoir 
de  Mecanique  pratique,"  page  328. 

The  experiments  of  Coulomb  upon  old  cords,  not  being 


RIGIDITY   OF   COEDS.  333 

otherwise  sufficiently  complete,  and  not  furnishing  any 
precise  data,  it  is  not  possible,  without  new  researches,  to 
give  a  rule  for  calculating  the  rigidity  of  these  cords. 

271.  Tarred  cards.  —  In  calculating  the  results  of  Cou- 
lomb's experiments  upon  tarred  cords,  as  we  have  done 
for  white  cords,  we  find  the  following  results  : 

w=30  threads  A=2.5312  B=0.041209 
7i=15  "  A=0.76701  B=0.019806 
n=  6  "  A=0.15341  B=0.0085293, 

which  differ  very  little  from  those  given  by  Navier.  But 
if  we  seek  the  resistance  answering  to  each  thread  we  find 


71=30  threads 

-=0.084376 
n 

5=0.0013736 
n 

71  =  15         " 

-=0.051134 
n 

?0.0013204 
n 

71=    6         " 

-=0.025568 

71 

-=0.0014215 
n 

Mean    .     .     0.0013718 

We  see,  by  this,  that  the  values  of  B  are  for  tarred  as 
for  white  cords,  sensibly  proportional  to  the  number  of 
threads,  but  it  is  not  so  with  the  values  of  A,  as  M.  N"a- 
vier  has  admitted. 

In  comparing,  as  we  have  done,  for  white  cords,  the 

* 

values  of  —  corresponding  with  the  cords  of  30,  15,  and 
n 

6  threads,  we  have  the  following  results  : 


334: 


RIGIDITY   OF   COEDS. 


TABLE. 


DIFFERENCES 

VALUES 

DIFFERENCES 

of  values  of 

NUMBER 

of 

DIFFEEENCE 

of  values  of 

A 

of 

A 

of  number  of  threads. 

A 

— 

threads. 

n 

n 

by  difference  of 

threads. 

30 

0.084376 

From  30  to  15,  15  threads 

0.033242 

0.00221 

15 

0.051134 

From  15  to  6,  9       do 

0.025566 

0.00284 

6 

0.025568 

From  30  to  6,  24    do 

0.058808 

0.00245 

Mean 0.00250 


It  follows  from  this  that  the  value  of  A  may  be  repre- 
sented by  the  formula 

A=n  [0.025568  +  0.0025  0—6)]  =  (0.010568  +  0.0025^, 

and  the  total  resistance  upon  a  roller  with  a  diameter  D  by 

K=^(O.Oi0568+0.002507i+0.001372Q)lbs- 

This  expression  has  the  same  form  as  that  for  white 
cords,  and  shows  that  the  rigidity  of  tarred  cords  is  a  lit- 
tle above  that  of  new  white  cords. 


272.  Table  of  the  rigidity  of  cords  of  different  diame- 
ters rolling  upon  a  drum  one  foot  in  diameter.  By  means 
of  the  two  formulae  deduced  from  the  experiments  of 
Coulomb,  for  new  white  and  dry  cords,  and  for  tarred 
cords,  the  only  ones  upon  which  experiments  anywise 
numerous  have  been  made,  we  have  formed  the  following 
tables,  for  which  we  have  calculated  approximately,  from 
the  data  of  Coulomb,  the  number  of  threads  answering  to 


EIGIDITY   OF  COEDS. 

different  diameters,  by  means  of  the  formulae 


335 


for  white  dry  ropes,  and 


for  tarred  ropes,  observing  that  the  cordage  of  6  threads 
of  Coulomb  appeared  to  be  too  small,  and  admitting  that 
the  numbers  of  threads  are  proportional  to  the  squares  of 
the  diameters. 


Table  of  rigidity  of  cords  rolled  upon  a  drum  one  foot  in 
diameter. 


Number  of 
threads. 

"WHITE  COEDS. 

TAERED  COEDS. 

Diameter. 

Value  of  the 
natural 
rigidity  A. 

Value  of  the 
rigidity 
proportional 
toQ. 

Diameter. 

Value  of  the 
natural 
rigidity  A. 

Value  of  the 
rigidity 
proportional 
toQ. 

ft. 

Ibs. 

ft. 

Ibs. 

6 

0.0294 

0.0767283 

0.0071458 

0.0346 

0.1534094 

0.00824488 

9 

0.0360 

0.1629580 

0.0107187 

0.0424 

0.2977124 

0.01236731 

12 

0.0416 

0.2810982 

0.0142916 

0.0490 

0.4870810 

0.01648975 

15 

0.0465 

0.4311487 

0.0178645 

0.0548 

0.7215150 

0.02061219 

18 

0.0509 

0.6131097 

0.0214375 

0.0600 

1.0010146 

0.02473463 

21 

0.0550 

0.8269810 

0.0250104 

0.0648 

1.3255796 

0.02885707 

24 

0.0588 

1.0727628 

0.0285833 

0.0693 

1.6952002 

0.03297951 

27 

0.0623 

]  .3504549 

0.0321562 

0.0735 

2.1099064 

0.03710194 

30 

0.0657 

1.6600575 

0.0357291 

0.0774 

2.5696680 

0.04122438 

33 

0.0689 

2.0015704 

0.0393020 

0.0813 

3.0744952 

0.04534682 

36 

0.0720 

2.3749938 

0.0428749 

0.0849 

3.6243878 

0  04946926 

39 

0.0749 

2.7803375 

0.0464478 

0.0884 

4.2193460 

0.05369169 

42 

0.0778 

3.2175717 

0.0500207 

0.0917 

4.8593698 

0.05771413 

45 

0.0805 

3.6867262 

0.0535936 

0.0949 

5.544459 

0.06183657 

48 

0.0831 

4.1877912 

0.0571666 

0.0980 

6.2746138 

0.06595901 

51 

0.0857 

4.720766 

0.0607395 

0.1010 

7.0498340 

0.07008145 

54 

0.0882 

5.2856523 

0.0643124 

0.1039 

7.87011984 

0.07420388 

57 

0.0906 

5.8824484 

0.0678853 

0.1068 

8.7354712 

0.07832632 

60 

0.0930 

6.511550 

0.0714582 

0.1096 

9.6458880 

0.08244876 

*  Morin  has  the  following  expressions  : 


rfcent.—  |/0.1338»i  for  white  cords, 
<£cent.—  j/o.  186/1  for  tarred  cords. 


336  KIGIDITT   OF   COEDS. 

273.  Moistened  cords.— As  to  wetted  cords,  the  results 
of  Coulomb's  experiments  are  not  sufficiently  conclusive 
for  the  establishment  of  any  practical  rule :  for  he  found 
that  for  cords  of  15  and  of  6  threads,  the  presence  of  water 
did  not  increase  the  rigidity,  and  that  for  a  cord  of  30 
threads,  the  constant  term  A,  representing  the  natural 
rigidity,  alone  was  increased,  and  that  nearly  twofold. 
M.  Navier,  and  other  authors  after  him,  admitted  in  this 
case,  that  it  was  necessary  to  double  the  value  of  the 
term  A,  while  preserving  the  same  value  for  the  term  B. 
But  upon  this  subject  we  need  new  experiments,  more 
complete  and  conclusive. 

274.  Use  of  the  preceding  tables  or  formula. — To  find 
the  rigidity  of  a  cord  of  a  given  diameter,  or  number  of 
strands,  we  first  obtain  in  the  table,  or  by  the  formula, 
the  value  of  the  quantities  A  and  B,  corresponding  with 
these  data,  and  knowing  the  tension  Q  of  the  enrolled 
strip,  we  shall  have  its  resistance  to  rolling  upon  a  drum 
one  foot  in  diameter  by  the  formula 

K^A-fBQ. 

Then  dividing  this  quantity,  by  the  diameter  of  the 
pulley  or  roller,  upon  which  the  cord  is  to  be  enrolled, 
we  shall  have  the  resistance  to  the  rolling. 

EXAMPLE. — What  is  the  rigidity  of  a  white  dry  cord, 
in  good  condition,  0.01918"-  in  diameter,  or  of  60  threads, 
which  rolls  upon  a  pulley  0.721 8ft-  diameter  at  the  groove, 
under  a  tension  of  1764.38  pounds  ?  The  table  gives  for 
the  white  dry  cord  in  good  condition,  for  60  threads  round 
a  drum  one  foot  in  diameter, 

A=6.51155    B=0.07U58. 
we  have 

D=0.7218ft-4-0.0918ft-=0.8136ft-, 


KIGIDITT   OF   COEDS. 


337 


and  consequently 


._1go 


0.8136 

The  total  resistance  to  be  overcome,  not  including  the 
friction  of  the  axles,  is  then 


We  see  in  this  example  that  the  rigidity  has  increased 
the  resistance  by  about  -  of  its  value. 


THE  DKAUGHT  OF  VEHICLES, 

AND   THE  DESTRUCTIVE  EFFECTS  WHICH   THEY   PRODUCE   UPON 
THE   ROADWAY. 

275.  The  draught  of  vehicles. — The  study  of  the  effects 
produced  by  the  motion  of  vehicles  may  be  divided  into 
two  distinct  parts :  the  draught  of  vehicles  proper,  and 
the  action  which  they  exert  upon  the  roadway. 

Researches  relative  to  the  draught  of  vehicles  have 
for  their  object  the  determination  of  the  intensity  of  the 
effort  to  be  exerted  by  the  motive  power,  according  to  the 
amount  of  the  load,  the  diameter  and  breadth  of  the 
wheels,  and  according  to  the  velocity,  and  the  condition 
or  nature  of  the  road. 

For  the  first  experiments  upon  the  resistance  expe- 
rienced by  cylindrical  bodies,  in  rolling  upon  each  other, 
we  are  indebted  to  Coulomb,  who,  on  the  occasion  of  his 
experiments  upon  the  rigidity  of  cords,  determined  the 
resistance  experienced  by  lignum  vitse  or  elm  rollers  upon 
plane  surfaces  of  oak,  placed  on  a  level. 

The  rollers  employed  being  placed  perpendicularly  to 
the  directions  of  the  pieces  of  oak,  flexible  twines  were 
passed  over  them,  at  each  end  of  which  was  suspended 
equal  weights,  and  according  to  the  number  of  cords  thus 
loaded,  the  total  pressure  was  made  to  vary. 

From  another  cord  passed  round  the  middle  of  the 
rollers  was  suspended  a  motive  weight,  whose  value, 


DRAUGHT   OF   VEHICLES. 


339 


experimentally  determined,  was  such  as  to  ensure  a  slow 
and  continuous  motion  approaching  uniformity. 

The  pressure  or  total  loads,  and  the  motive  weights  de- 
termined by  the  experiments  of  Coulomb,  are  given  in 
the  following  table : 


Nature  of  rollers. 

Pressure. 

Resistance  for  the  diameters 

of  6.55  in. 

of  2.18  in. 

Lignum  vitae  
Elm... 

i 

220.54 
1102.74 
2205 
diameter 
2205 

1.32 
6.62 
13.23 
of  13.11  in. 
11.02 

3.52 
20.73 
39.69 
of  6.55  in. 
22.05 

An  examination  of  the  results  of  these  experiments 
shows  that  the  resistance  was  sensibly  proportional  to  the 
pressure,  and  in  the  inverse  ratio  of  the  diameter  of  the 
rollers. 

Similar  experiments  have  been  made  at  Yincennes, 
and  at  the  Conservatory  of  Arts  and  Manufactures,  with 
wooden  rollers  of  different  diameters  rolling  upon  wood, 
leather  and  plaster.  The  body  of  the  rollers  was  always 
nearly  0.656ft-  in  diameter,  and  the  method  of  observation 
similar  to  that  of  Coulomb's.  Only  the  total  spaces  were 
greater,  and  the  movements  were  observed  with  more 
accurate  means. 

According  to  the  character  of  the  apparatus,  if  we  call: 

Q  the  motive  weight,  which  in  each  case  maintains  a 
uniform  motion ; 

T  the  radius  of  the  part  of  the  roller  which  represents 
the  wheel ; 

/  the  radius  of  the  body  of  the  roller,  or  arm  of  lever 
of  the  motive  weight ; 

R  the  resistance  to  rolling  ; 

We  have  during  uniform  motion  the  relation 


34:0  DRAUGHT   OF   VEHICLES. 

from  whence  we  deduce  in  each  case 


K=Q-. 

T 


According  to  the  law  admitted  by  Coulomb,  the  re- 
sistance to  rolling  being  proportional  to  the  pressure, 
which  we  call  P,  and  in  the  inverse  ratio  of  the  radius  or 
diameter  of  the  rollers,  may  be  expressed  by  the  formula 


in  which  A  is  a  constant  number  for  each  kind  of  earth, 
but  variable  for  different  kinds  and  different  conditions 
of  the  same  kind. 

"We  will  give  here  some  results  of  experiments  made 
at  Yincennes  and  at  the  Institute,  as  well  as  all  the  data, 
by  means  of  which  we  have  calculated  for  each  case  the 
value  of  the  number 


which  should  be  constant  according  to  the  law  of  Coulomb. 

These  results,  inscribed  in  the  following  table,  prove 
that  the  law  of  Coulomb  still  applies,  with  all  desirable 
exactness  for  practice,  and  furthermore  that  the  resistance 
increases,  when  the  width  of  the  parts  in  contact  is  dimin- 
ished. 

Other  experiments  of  the  same  kind  have  confirmed 
these  conclusions,  and  we  must  admit,  at  least  as  suffi- 
ciently exact  laws  of  practice,  that  for  wood,  plaster, 
leather,  and  generally  for  all  hard  bodies,  resistance  is 
very  nearly  : 

1st.  Proportional  to  the  pressure. 

2d.  And  is  in  the  inverse  ratio  of  the  diameter  of  the 
rollers. 


DEATJGHT   OF  VEHICLES. 


341 


3d.  And  is  so  much  the  greater,  as  the  width  of  the 
zone  of  contact  is  the  smaller. 


Experiments  on  oak  rollers,rolling  upon  poplar. 


Widths  of 
strips  of 

Pressure 
of  rollers 

Motive 
weight 

Leverage 
of  motive 

Leverage 
of 

Value 
of 

Value  of  the 
number 

poplar. 

P. 

oj 

weight 

resistance 
r. 

resistance 
E. 

F' 

ft. 

Ibs. 

Ibs. 

ft. 

ft. 

Ibs. 

C 

435.67 

3.862 

0.3296 

0.59384 

2.1437 

0.002922 

ft  <19£     J 

386.31 

3.494 

0.3190 

0.44456 

2.4789 

0.002852 

v.oZo     -\ 

370.54 

2.967 

0.3329 

0.29593 

3.3391 

0.002666 

( 

409.59 

3.781 

0.3346 

0.14764 

8.4867 

0.003058 

Mean... 

0.002874 

/- 

440.77 

7.861 

0.3307 

0.59384 

4.362 

0.005880 

0.081    \ 

374.61 

7.034 

0.3329 

0.29593 

7.913 

0.006253 

I 

413.96 

8.137 

0.3313 

0.14764 

19.602 

0.006991 

Mean... 0.006374 


276.  Experiments  upon  vehicles  moving  upon  common 
roads. — The  experiments,  some  results  of  which  we  have 
just  cited,  do  not  suffice  to  authorize  the  extension  of 
these  conclusions  to  the  movement  of  vehicles  upon  com- 
mon roads.  It  is  necessary  to  operate  directly  upon  the 
vehicles  in  the  usual  circumstances  of  their  service.  Ex- 
periments on  this  subject  were  first  undertaken  at  Metz 
in  1837  and  1838,  then  at  Courbevoi  in  1839  and  1841, 
with  vehicles  of  all  kinds,  and  a  separate  study  was  made 
of  the  influence  of  the  pressure,  that  of  the  diameter  of 
the  wheels,  that  of  their  width,  that  of  the  velocity  of 
translation,  and  that  of  the  condition  of  the  ground  upon 
the  intensity  of  the  draught. 

To  indicate  the  course  followed  in  the  discussion  of 
the  immediate  results  of  experiments,  we  will  call : 

P  the  whole  weight  of  the  vehicle,  exclusive  of  the 
wheels ; 


34:2  DRAUGHT  OF  VEHICLES. 

pr  andy  the  respective  weights  of  the  fore  and  hind 
wheels  ; 

Pj  the  total  pressure  upon  the  ground  ; 

P'  and  P"  the  components  of  the  weight  P  upon  each 
of  the  axles  ; 
r'  and  r"  the  radii  of  the  fore  and  hind  wheels  ; 

/•/  and  r"  the  mean  radii  of  each  of  the  boxes  of  the 
wheels  ; 

F  the  effort  of  traction  in  the  direction  of  the  shafts  ; 

F4  the  component  of  this  effort  parallel  to  the  ground. 

The  pressure  of  the  forward  part  of  the  vehicle  upon 
the  ground  will  be  P'+y,  that  of  the  hind  part  V'+p", 
and,  according  to  the  law  of  Coulomb,  the  resistance  to 
rolling  will  be  : 


For  the  fore  part  of  vehicle  A  .  . 

T 

~PfrJL.^ff 

For  the  hind  part  of  vehicle  A  .  —  -5^—. 

T 

' 

The  friction  of  the  boxes  against  the  axles,  referred  to 
the  circumference  of  the  wheel,  will  be 


For  the  fore  part/.  :~ 

// 

For  the  hind  party. 


. 

Finally,  if  the  ground  slopes,  the  component  of  the 
total  weight  will  be 


calling  H  the  difference  of  level  corresponding  with  the 
length  L.  This  component  will  moreover  act  as  a  resist- 
ance in  ascents,  and  as  a  power  in  descents. 


DRAUGHT   OF   VEHICLES. 


343 


From  this  it  is  evident  that,  in  the  ascent,  for  exam- 
ple, the  total  resistance  .  to  rolling  will  be  in  calling  R' 
and  R"  the  resistances  of  the  fore  and  hind  wheels 


(PV  ' 


The  quantities/,  F  P",  /,  T",  r,f,  *•/',  F,y,y,  H  and  L 
have  been  given  by  direct  measures,  the  quantity  Fj  by 
experiments  made  with  the  dynamometer  ;  we  have  had 
thus  for  each  case  the  value  of  R'+R'^R. 

If  the  law  of  Coulomb  is  true,  we  have  further 


whence  we  may  derive  the  value  of  the  quantity  A, 
which  should  be  constant, 


A  


R 


~~~ 


If  the  wheels  are  equals,  the  above  expression  is  re- 
duced to 

Hr  TLr 


277.  Ratio  of  the  draught  to  the  load.  —  We  would 
here  remark  that  if  the  law  of  Coulomb  is  admitted,  the 
horizontal  effort  of  traction  to  be  exerted  upon  level 
ground  will  have,  from  what  precedes,  for  its  expression, 


F,=A 


344  DRAUGHT   OF  VEHICLES. 

or,  since  Hie  radii  of  the  boxes  are  usually  the  same 
for  the  fore  and  hind  wheels,  and  equal  to  a  mean 
value  0*,, 


We  see  also  that  if  we  would  make  the  draught  for  the 
fore  and  hind  wheels  the  same,  we  must  make 


a  condition  which,  by  reason  of  the  near  relation 


reduces  to 

p/^-p,., 

r'     r"' 

In  other  words,  we  must  so  arrange  it  that  the  load  shall 
be  distributed  in  the  direct  ratio  of  the  radii  of  the  wheels  ; 
but  we  shall  see  that  the  transportation  business,  which 
nearly  conforms  to  this  rule  in  practice,  has  for  its  inter- 
est an  increase  of  the  load  upon  the  great  wheels,  beyond 
the  proportion  to  which  we  have  been  conducted. 

If  we  admit  this  proportion,  and    remember    that 
P'+P"=P,  we  shall  find  that 


since  we  have  very  nearly 


F_FX_     P 
r/~V/~V+r" 


DRAUGHT   OF   VEHICLES.  345 

If  we  now  seek  the  value  of  the  ratio  of  the  draught 
to  the  load,  we  shall  find 

o..»#M 

/     '  T>    '  i5  * 


r'+r 


In  the  application  to  wagons  heavily  loaded,  which 
are  the  most  important  for  our  consideration,  the  weight 
of  the  wheels  is  but  a  very  small  fraction  of  the  load,  and 
the  weight  proper  of  the  body  of  the  wagon,  and  it  may 
be  neglected  alongside  of  the  total  load,  which  reduces 
the  ratio  to 

w=     /      ,,     f°r  vehicles  with  four  wheels,  or  to 

jTj         T  -\-T 

~=      J  l  for  vehicles  with  two  wheels. 
*i«       T 

These  experiments  will  hereafter  serve  to  determine 
by  experiment  the  ratio  of  the  draught  to  the  load  for  the 
most  usual  cases. 


278.  Influence  of  the  pressure. — To  ascertain  the  influ- 
ence of  the  pressure  upon  the  resistance  to  rolling,  we  set 
in  motion  the-same  vehicle  with  different  loads,  upon  the 
same  road,  in  the  same  condition.  We  give  the  results 
of  some  of  these  experiments  made  at  a  walking  pace. 


346 


DRAUGHT   OF   VEHICLES. 


TABLE. 

Experiments  upon  the  influence  of  pressure  upon  the 
draught  of  vehicles. 


Vehicles  used. 

The  routes  run  over. 

Pressure. 

Draught 

Ratio  of 
draught  to 
load. 

Ibs. 

Ibs. 

1 

13215 

398.4 

QQ  1 

Artillery  Ammu- 
nition wagon. 

Road  from  Courbevoie  to 
Colombes,  dry  in  good 
order  and  dusty. 

13541 

352.6 

dO.  J. 
1 

"381 
1 

10101 

250.7 

40^2 

1 

15716 

306.3 

5T.3 

Wagon  without  , 
springs. 

Road  from  Courbevoie  to 
Bezons,  solid,  hard 
gravel,  very  dry. 

12037 
9814 

245.9 
205.5 

1 

1&9 

1 

47.7 

1 

7565 

150.8 

50.1 

1 

1 

From  Colombes  to  Cour-  1 

3528 

86.6 

40.8 

Wagon  with 
springs. 

bevoie,  paved,  in  good  ( 
order,  with  wet  mud. 

7260 

196.7 

1 
36.9 

1 

11018 

299.9 

"3678 

" 

1 

6616 

306.3 

21.6 

Wagon  with  six 

1 

equal  wheels. 
Two  wagons  con-  - 
nected  with  six 

?rom  Courbevoie  to  Co- 
lombes; deep  ruts;  wet 
detribus. 

10348 

494.0 

~2l7 
1 

equal  wheels. 

13232 

630.3 

TiT 

1 

13232 

632.3 

~2l7 

It  results  from  an  examination  of  this  table  that  upon 
solid  metalled  roads,  and  upon  pavements,  the  resistance 
to  the  draught  of  wagons  is  sensibly  proportional  to  the 
pressure. 


DRAUGHT   OF   VEHICLES.  347 

We  would  observe  that  the  experiments  made  with 
one  only  or  two  wagons  with  six  wheels  have  given  the 
same  total  draught  for  the  same  load  of  16232  pounds, 
vehicles  included.  It  follows  from  this,  that  the  draught 
is,  all  else  being  equal,  and  moreover  within  certain 
limits,  independent  of  the  number  of  wheels.  We  may 
also  draw  the  same  consequence  from  results  given  in 
the  following  table,  relative  to  the  same  wagon  employed 
successively  with  six  and  four  wheels.  The  resistance 
was  the  same  in  the  two  cases  for  the  same  load. 

279.  Influence  of  the  diameter  of  the  wheels. — To 
study  the  influence  of  the  diameter  of  wheels  upon  the 
draught,  we  have  respectively  set  running  over  the  same 
part  of  the  road,  in  the  same  condition,  wagons  having 
the  same  weight,  and  having  equal  widths  of  tires,  but 
with  diameters  differing  in  very  extended  limits.  We 
publish  in  the  following  table  some  of  the  results  ob- 
tained. 

We  have  also  compared  with  the  artillery  ammunition 
wagon,  whose  wheels  have  a  diameter  of  6.654:ft-,  different 
kinds  of  vehicles,  including  drays  whose  wheels  had  a 
diameter  not  above  from  1.94:2ft-  to  1.378ft-  The  ratio 
of  the  draught  to  the  pressure  varied  from  ^  for  the 
largest  wheels  to  •£•%  for  the  smallest,  upon  a  road  paved 
with  sandstone  from  Fontainbleau.  Upon  the  metalled 
road  from  Courbevoie  to  Colombes,  the  experiments  were 
made  upon  diameters,  comprised  between  6.654ft-  and 
2.86ft- 

These  examples  show  that  upon  solid  roads  we  may 
admit,  as  a  law  for  practice,  that  the  draught  varies  in  the 
inverse  ratio  of  the  diameter  of  the  wheels. 


348 


DRAUGHT   OF   VEHICLES. 


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DRAUGHT   OF   VEHICLES.  349 

General  Piobert,  who  gave  his  attention  to  theoretical 
and  experimental  researches  upon  the  resistance  to  rolling, 

concluded  that  this  resistance  varied  in  the  inverse  ratio 

2 

of  a  power  of  the  diameter  included  between  -  and  unity, 

o 

approaching  more  nearly  to  the  last  limit  as  the  ground 
is  harder ;  and  that  upon  pavement  this  resistance  varies 
in  the  inverse  ratio  of  the  radius  of  the  wheel,  increased 
by  the  roughness  of  the  pavement. 

If  we  would  apply  this  law  to  experiments  made  upon 
roads  of  solid  metalling,  dry  or  moist,  admitting,  for  ex- 
ample, that  the  power  of  the  radius  to  be  employed  is 
4. 

-,  we  find  the  results  inserted  in  the  last  column  of  the 
5 

preceding  table.  An  examination  of  these  results,  which 
are  expressed  in  common  fractions,  shows  that  the  law  of 
Coulomb  represents  the  results  of  an  experiment  made 
upon  the  road  from  Courbevoie  to  Colombes  with  an 

exactitude  of  - —  ;  while  by  varying  the  resistance  in  the 
12.5 

4. 

inverse  ratio  of  the  power  -  of  radius  we  obtain  an  ap- 

5 

proximation  of  — . 
15 

£Tow,  in  such  researches,  we  seldom  obtain  direct 
results  of  experiment,  which  do  not  differ  more  than  from 

—  to  -— ,  corresponding  to  the  limits  calculated  by  these 
12  15 

two  laws.  It  follows,  therefore,  that  in  practice,  we  may 
for  solid  roads  adopt  the  simple  law  of  Coulomb,  without 
fear  of  committing  a  grave  error. 

280.  Influence  of  the  width  of  the  rims. — This  influ- 
ence was  first  investigated  with  an  apparatus  composed 
of  a  cast-iron  axle,  upon  which  were  placed  cast-iron  discs 
turned  at  the  periphery,  and  forming  at  once  the  load 


350 


DRAUGHT   OF   VEHICLES. 


and  the  wheels,  whose  total  width  was  thus  proportional 
to  their  number ;  subsequently  they  used  common  wagon 
wheels,  having  the  same  diameter,  but  unequal  breadths. 
Some  of  the  results  of  these  experiments  are  recorded  in 
the  following  table : 

Experiments  upon  the  influence  of  the  widths  of  felloes 
upon  the  resistance  to  rolling. 


1 

•s 

-  2 

J* 

*. 

Vehicles  employed. 

Ground  passed  over. 

Diameter  of 

3  £ 

I  S  pT 

.2  "g  PS 

•s 

wheels. 

P* 

ft 

1 

Polygonal  enclosure  at  / 

ft 
2.5827 

ft. 

ft. 
0.1476 
.2953 

Ibs. 
2298.1 
2944.3 

Ibs. 
853.3 
461.4 

0.1982 
.2023 

Apparatus  with 
cast-iron 
axle. 

Metz.                f 

Manoeuvring   shed    at 
the  Metz  school,  sand 
from  0.39ft.  to  0.49ft 

2.5827 

f 

.4429 
.1476 
.2958 
.4429 
.6069 

3191,3 
2306. 
2988.4 
8178.3 
3043.5 

861.2 
556.2 
589.5 
597. 
448.1 

.1603 
0.3114 
.2547 
.2425 
.2070 

i 

.7382 

8671. 

571.8 

.2011 

Artillery  ammuni- 

Eoad from  Courbevoie 

4.7179 

4.7179 

.5741 

7639.7 

166.5 

0.0514 

tion  wagon. 

to  Colombes,  wet 

4.6458 

4.6458 

.1968 

7957.3 

166.3 

.0496 

Artillery  ammuni- 
tion wagon. 

"Wagon  6  wheels- 

Paved  with  sandstone 
of                  } 
Fontainbleau. 

4.7179 
4.7671 
4.7671 
2.8215 

4.7179 
4.7671 
4.7671 
2.8215 

.5741 
.8773 
.8773 
.1968 

12165.4 
12169.8 
10131.9 
7211.9 

183.0 
160.1 
128.4 
157.9 

.0355 
.0314 
.0302 
.0309 

These  examples  show : 

1st.  That  upon  soft  foundations,  the  resistance  in- 
creases as  the  width  of  the  tire  decreases,  and  for  farming 
purposes  we  should  use  tires  of  alout  0.33ft-  width. 

2d.  That  upon  solid  roads,  metalled  or  paved,  the  re- 
sistance is  nearly  independent  of  the  width  of  the  tire. 

281.  Influence  of  the  velocity. — To  appreciate  the 
influence  of  the  velocity  upon  the  draught  of  vehicles  we 
have  put  in  motion,  upon  different  roads  in  different  con- 
ditions, the  same  vehicles,  changing  in  each  series  of 
experiments  only  the  velocity,  which  was  successively, 
that  of  a  walk,  a  fast  walk,  a  trot,  and  a  smart  trot.  Some 


*  These  values  of  A  are  for  the  unit  of  foot  of  radius. 


DRAUGHT   OF  VEHICLES. 


351 


of  the  results  of  these  experiments  are  reported  in  the 
following  table : 

Experiments  upon    the  influence  of  velocity  upon    the 
resistance  to  the  draught  of  vehicles. 


Vehicles  used. 

Ground  run  over. 

Load. 

Gait 

1 

Draught. 

Eatioof 
draught 
to  load. 

Ibs. 

ft. 

Ibs. 

1 

" 

*< 

walk 

4.59 

363.9 

6.3 
1 

trot  

9.19 

370.5 

— 

Apparatus 

Polygonal  en- 

6.2 

with  cast- 
iron  axle. 

closure  at  Metz, 
wet  and  soft. 

2944.3  \ 

walk 

4.19 

474.18 

1 

6.21 
1 

( 

trot  

11.09 

434.48 

(Trr 

41  O 

i 

• 

- 

' 

walk  

.13 

202.90 

40.76 

Road  from  Metz 

1 

Carriage  No. 
16,  with  its" 
piece. 

to   Montigny, 
metalled,  very 
smooth,     and   " 
dry. 

8270.5  - 

fast  walk 
trot  

4.99 

8.04 

202.90 
224.96 

40.76 
1 
36.76 

1 

• 

- 

• 

quick  trot 

12.40 

396.99 

30.99 

1 

} 

r 

walk  

4.07 

317.59 

22.83 

" 

1 

Stage  wagon 

Paved     with 

fast  walk 

5.57 

337.44 

21.49 

hung  on  six 

sandstone   from  }- 

7251.6  - 

1 

springs. 

Fontainbleau. 

trot  

7.74 

355.08 

20.42 

1 

; 

J 

- 

quick  trot 

11.81 

382.65 

iaw 

We  see,  by  these  examples,  that  the  draught  does  not 
increase  sensibly  with  the  velocity^  upon  soft  bottom,  but 
that  upon  solid  and  uneven  roads,  it  increases  with  the 
inequalities  of  the  ground,  the  stiffness  of  the  wagon,  and 
the  rapidity  of  the  motion. 


282.  Approximate  expression  for  the  increase  of  resist- 
ance with  the  velocity. — To  ascertain  the  relation  existing 


352  DRAUGHT   OF   VEHICLES. 

between  the  resistance  and  the  velocity  upon  hard  and 
uneven  ground,  we  have  taken  the  velocities  for  the  ab- 
scissa, and  for  the  ordinates,  the  values  of  the  number  A, 
furnished  by  experiment,  and  this  graphic  representation 
of  the  results  has  shown  that  all  the  points  so  determined 
were  for  each  series  of  experiments  situated  very  nearly 
upon  the  same  straight  line.  Thus  the  experiment  rela- 
tive to  a  battery  carriage,  loaded  with  its  piece,  a  very 
rigid  carriage,  moved  at  different  velocities  upon  a  very 
well  metalled  road,  and  upon  the  pavement  of  the  city  of 
Metz,  are  represented  in  Fig.  109,  IY.  and  Y.,  and  we  see 
that  the  value  of  the  ordinates,  or  of  the  number  A,  in- 
creased with  the  abscissa  or  velocities  according  to  a  law 
which,  in  the  limits  of  experiment,  may  be  expressed  by 
a  straight  line  cutting  the  axis  of  ordinates  at  a  certain 
height,  which  indicates  that  for  a  velocity  zero,  the  resist- 
ance has  still  a  certain  value,  or  generally  is  composed  of 
one  part  independent  of  the  velocity,  and  of  one  part  pro- 
portional to  it,  this  resistance,  or  rather  the  value  of  the 
number  A  may  then  be  generally  represented  by  an  ex- 
pression of  the  form 


in  which 

a  is  a  constant  number  for  each  kind  of  road-bed,  in  a 
determined  condition,  and  which  expresses  the  value  of 
tiie  number  A  for  a  velocity  Y=l  metre  =3.2809fti,  which 
is  that  of  a  gentle  walk  ; 

d  is  a  constant  factor  for  each  kind  of  road-bed  and  of 
carriage. 

In  the  particular  case  of  the  two  series  of  experiments 
above  cited,  we  have  for  a  battery  carriage  with  its  gun 

Upon  the  road  to  Montigny  jj^         j^          ft> 

very  well  metalled  ...........................  A=0.010  +  0.002  (0.3047  V—  1). 

Upon  the  Pavement  of  Metz  l^s         j^s          ft> 

of  sandstone  from  Sierck  ...................  A=0.0066  +  0.0057  (0.3047  V—  1). 


DRAUGHT   OF    VEHICLES*  353 

These  examples  suffice  to  show : 

1st.  That  at  a  walk,  the  resistance  upon  good  pave- 
ment is  less  than  upon  a  very  good  metalled  road, 
quite  dry; 

2d.  That  with  good  speed,  the  resistance  upon  pave- 
ment increases  rapidly  with  the  velocity  Y. 

Thus  at  a  walk,  upon  the  pavement  of  Metz,  with 
wheels  3.28ftj  radius,  the  resistance  would  be  6.6lbs-  for 
every  1000lbs-  of  the  total  load,  vehicle  included,  while  at  a 
smart  trot,  V=4m-=13.12ft-,  it  would  be 


Ibs. 


6.6lbs--h5.7lbs-x  3=23.7 

that  is  to  say,  nearly  fourfold. 

Upon  the  pavement  of  Fontainbleau,  with  wide  joints, 
and  rounded  edges,  which  offers  many  inequalities  whose 
elements  may  be  displaced  under  the  action  of  the  load, 
the  resistance  at  a  walk  is  much  greater  than  upon  the 
pavement  of  Metz,  and  the  increase  of  the  resistance  with 
that  of  the  velocity  is  still  more  rapid.  Fig.  VI.,  which 
represents  the  results  of  the  experiment  obtained  with  a. 
wagon  of  the  general  coach  establishment,  whose  springs 
were  wedged  up,  shows  that  the  inclination  of  the  straight 
line  or  the  increase  of  the  velocity  is  much  greater  than 
upon  the  pavement  of  Metz,  and  we  deduce  from  it,  for 
the  representation  of  the  value  of  A,  the  formula 

A=0.0092lb9-+0.0089lb9- (0.3047  Yft—l), 

which  shows  that  for  wheels  of  3.28ft-  radius  the  resistance 
at  a  walk  of  3.28ft-  velocity  will  be  9.2lbs-  for  every  thou- 
sand pounds  of  load ;  that  is  to  say,  nearly  one-half  above 
that  upon  the  pavement  of  Metz,  and  for  a  smart  trot  at  a 
velocity  4m  =13.12ft-  in  V  it  will  be  equal  to,  for  every 
1000  pounds, 

9.2lbs-+8.9lbs-x3=35.9lb% 

while  that  upon  the  pavement  of  Metz.  was  only  23.7lb8> 
23 


354  DRAUGHT   OF    VEHICLES. 

As  for  wagons  on  springs,  experiment  shows  that  the 
resistance  also  increases,  but  much  slower  with  the  ve- 
locity upon  roads  with  an  uneven  surface.  Thus  upon 
the  pavement  of  Paris  (Fig.  VII.),  the  same  stage  wagon, 
whose  springs  were  restored  to  free  action,  has  given  only 
for  the  value  of  A  the  expression 

Ar=:0.0098lbs-+0.0025lbs-  (.3047  V-l), 

so  that  for  a  trot,  at  the  velocity  of  4.  metres,  and  wheels 
lm-  radius,  the  resistance  for  a  load  of  1000  pounds  would 
be  but 

9.8lbs-+2.5lbs-  x3=17.30lb% 

that  is  to  say,  one-half  of  that  experienced  by  the  same 
wagon,  unhung,  upon  the  same  pavement  and  with  the 
same  velocity. 

283.  Practical  consequences  of  these  experiments. — 
These   experiments   show,  on   the   one   hand,  the  great 
advantage»with  respect  to  traction  and  economy  of  motive 
power  possessed  by  wagons  with  springs  over  those  with- 
out them,  and  on  the  other  hand,  the  superiority  of  pave- 
ments with  narrow  tight  joints  and  smooth  surface,  over 
those  with  wide  joints  and  uneven  surface,  generally  used 
in  Paris.     These  results,  obtained  in  1837  and  published 
in  1838,  attracted  the  attention  of  engineers,  and  we  may 
suppose  have  stimulated  trials,  which  have  since  been  suc- 
cessful, for  the  use  of  cut  stone  blocks  of  regular  forms, 
whose  advantages  the  public  can  easily  appreciate. 

284.  Comparison  of  paved  and  metalled  roads. — The 
same  experiments  show  us  that,  if,  for  rolling  at  a  walk, 
paved  roads  offer  an  advantage  over  the  metalled,  it  is 
not  so  for  great  velocities  upon  good  metalled  roads,  dry, 
and  in  perfect  order;  but  that  when  these  roads  are  wet, 
the  pavement  resumes  its  advantage.     In  fact,  we  find  for 


DRAUGHT   OF  VEHICLES.  355 

this  last  case,  that  upon  the  road  from  Metz  to  Nantz, 
wet,  with  some  mud  and  pebbles  upon  a  level,  the  value 
of  the  number  A,  which  represents  the  resistance  for  1000 
pounds  of  load,  with  wheels  3.28ft-  radius,  for  the  dili- 
gences of  the  general  stage  department,  with  springs,  is 
given  by  the  formula 

A=0.014:lbs-  +  0.0022lbs-  (0.3047  V— 1). 

Comparing  this  with  that  obtained  for  the  pavements  of 
Paris,  we  find  that  the  draught  per  1000  pounds  with 
wheels  3.28ft>  radius  would  be : 

ft.        ft.        ft.         ft. 

For  velocities  of. 3.28      8.20      9.84      13.12 

Ibs.  Ibs.        Ibs.          Ibs. 

Upon  the  wet  metalled  road  from  Nantz H.         17.30     18.40      20.60 

Ibs.  Ibs.        Ibs.         Ibs. 

Upon  the  pavement  of  Paris 9.8      13.55     14.80      17.30 

The  excess  of  draught  experienced  upon  wet  metalled 
roads,  arises  chiefly  from  their  compressibility ;  and  natu- 
rally increases  in  proportion  to  the  softness  of  the  mate- 
rials, the  moisture  of  the  road,  and  to  a  want  of  proper 
maintenance. 

This  last  circumstance  has  upon  the  resistance  to  trac- 
tion a  great  influence,  whose  consequences — injurious  to 
the  economy  of  transportation — have  not  met  with  a 
proper  attention.  Experiments  made  in  September  and 
October,  1841,  with  the  same  wagon,  running  successively 
over  various  parts  of  the  same  road,  show,  the  mate- 
rials and  season  being  the  same,  the  draught  of  this  wagon, 

upon  portions  in  good  condition,  to  have  been  from  —  to 

do 

—  of  the  load,  while  on  the  parts  badly  managed  it  rose 

OD 

from  A to  k- 


356  DEATJGHT  OF  VEHICLES. 

285.  Influence  of  the  inclination  of  the  traces.  —  To 
study  the  influence  of  this  element  of  the  question,  we 
made  use  of  a  battery  carriage  with  equal  wheels,  the 
pole  being  inclined 

1°35',  3°35',  6°30',  8°30',  11°,  and  13°30', 

and  set  this  carriage  in  motion  upon  ground  covered  with 
wet  grass,  preserving  otherwise  the  same  weight  and  the 
same  velocity  in  all  cases. 

The  effort  of  traction  F,  measured  by  the  dynamome- 
ter, and  exerted  in  the  direction  of  the  traces,  is  evidently 
resolved  into  two  forces,  the  one  F'  horizontal  and  par- 
allel to  the  ground,  which  produces  the  motion,  and  sur- 
mounts all  the  resistances,  the  other  vertical,  F",  which 
diminishes  the  pressure  of  the  fore  wheels  upon  the 
ground. 

From  this  it  follows,  in  preserving  the  notations  of 
No.  276,  that  the  pressure  upon  the  ground  may  be  ex- 
pressed by 

0.96  [P'+P"]  +0.4F. 

So  that  calling 

f  the  ratio  of  the  friction  to  the  pressure, 

T\  the  mean  radius  of  the  boxes, 

T  that  of  the  wheels, 

L  the  total  space  run  over, 

The  equation  of  motion  of  this  carriage,  upon  level 
ground,  was  approximately 


Moreover,  we  have 

^=0.124"-,  r=2.565ft-,  /=0.065. 
Now,  before  going  further,  we  will  observe  that  on 


DBAFGHT  OF  VEHICLES.  357 

account  of  the  smallness  of  the  term 


we  may  evidently  neglect  the  value  of  this  expression, 
and  reduce  that  of  R'-f-B",  derived  from  the  preceding 
equation,  to 


F'--  L  (p'+p"_ 

r 

=F'—  .00303  (P'-fP"—  F"), 
and,  on  the  other  hand,  we  know  that 


we  have  then  to  compare  the  results  of  the  above  formula 
with  experiments,  the  relation 

.  _F/_  0.00303  (P/+P//--F//) 
P.-F" 


Now,  this  comparison  has  given  the  following  results, 
which  are  the  means  of  many  experiments  repeated  for 
each  case : 

Inclination  of  draught  ....1°35'  I     3°35'  I     6°30'    I     8°30'  I        11°  I   13°30' 
Value  of  uumber  A*  ....0.1145  j  0.1145  [  0.11778  |  0.1145  |  0.1145  |  0.1017 

The  agreement  of  all  these  values  shows  that  the  me- 
chanical effects  take  place  exactly  as  indicated  by  the 
formula,  when  account  is  taken  of  the  resolution  of  the 
efforts;  consequently,  to  ascertain  the  inclination  an- 
swering to  the  maximum  of  effect,  we  find,  by  known 
methods  of  calculation,  that  calling  h  the  height  of 
the  forward  point  of  attachment  of  the  traces,  above 

*  The  value  of  A  is  for  a  wheel  one  foot  radius. 


358  DRAUGHT   OF   VEHICLES. 

that  of  the  hind  attachment,  Id,  the  horizontal  projection 
of  the  distance  of  these  two  points,  the  ratio  of  the  quanti- 
ties answering  to  the  maximum  effect  of  the  motive  power 
will  be 


~b~  r—OAfrl  ' 

This  expression  shows  that  for  a  given  wagon,  the  in- 
clination of  the  traces,  or  the  value  of  j  increases  with 

that  of  A  or  the  resistance  of  the  road  bed,  and  in  this 
respect  the  site  chosen  for  the  experiment  was  very  suita- 
ble for  the  purpose.  Moreover,  for  the  same  ground,  the 
inclination  increases  as  the  radii  of  the  wheels  are  dimin- 
ished. 

Applying  the  above  relation  to  the  battery  carriage 
and  to  the  ground  of  the  experiment,  for  which  we  have 


-,  A=0.1145, 
we  find 


20.9' 

If  we  had  with  the  same  data  ^=0.820ft,  as  for  drays, 
we  should  have 


6.71' 

a  quantity  much  smaller  than  that  generally  in  use. 

Upon  metalled  roads,  for  which  A =.0492,  with  the 
usual  state  of  moisture  and  maintenance,  we  should  find 
for  artillery  wagons 

^=0.022=i 


which  is  nearly  the  inclination  adopted  for  battery  car- 
riages designed  for  long  marches. 


DEACJGHT   OF   VEHICLES.  359 

It  is  not  worth  while  to  extend  this  discussion,  to  which 
constructors  generally  attach  more  importance  than  it 
deserves,  and  we  limit  ourselves  to  saying  that,  within 
the  limits  where  it  is  necessarily  constrained,  the  inclina- 
tion of  the  traces  has  but  little  influence  upon  the  draught, 
and  that,  in  common  cases,  it  must  be  very  slight. 

286.  Recapitulation  and  application  of  the  general 
results  of  experiments. — The  following  table  gives  the 
values  of  the  ratio  of  draught  to  the  load,  for  a  great  num- 
ber of  different  circumstances,  the  formula  of  No.  277, 
combined  with  the  direct  results  of  experiments,  enabled 
us  to  calculate  approximately  the  value  of  this  ratio,  for 
the  usual  proportions  of  wagons  employed  in  trade. 


360 


DRAUGHT   OF  VEHICLES. 


STATEMENT  OF  THE  DEAUGHT  AND  THE  LOAD  OF  CAEEIAGE8, 


Artillery  carriages 

Artillery  wagons. 

Comtois1'  wagons. 

and  carts. 

W=0.229ft  to  0.246ft 

W=0.196ft.  to  .229ft. 

)esignation  of  the  route 
passed  over  by  the 
carriage. 

W=0.32ft.  to  0.89ft 
rl=0.124ft. 
r'=2.565ft. 
r'+r"=5.130ft 

r1=0.124ft. 
r'=1.885ft. 
r"=2.558ft 
r'+r"=4.4426ft 

r1=0.087ft 
r'=2.049ft. 
r"=2.377ft 
r'+r"=4426ft. 

/r1=.00806 

//•i=.00806 

/r1=.00574 

Earth  driftway  in  good 
order,  nearly  dry  

1      ] 
84:8 

1 

soil 

1 
81 

Solid  driftway,  with  bed 
of  gravel  .09ft  to  .13ft. 

1 

1 

1 

thick                

11.9 

Solid  driftway,  with  bed 

.of  gravel  .16  ft.  to  0.19ft. 

1 

1 

1 

tt 
Fin 
el 

lick               

11.6 

loii 

iol 

n  ground,  with  grav- 
bed,  0.33  to  0.49  ft 

new  road      

i 

Driftway    with    untrod 
snow      

i 

1  fi  A. 

i 

i  a 

i 

Firm  earth  bed  of  fine 

JUUI 

lo 

i 

sand,  with  gravel  0.33 

to  n  J.Q  ft.  thir-t 

m2 

8iT 

8i9      - 

in  good  order,  very 
dry  and  smooth.. 

I  walk   g2J 
jtrot     JL 

1 
54.3 

1 

[           50.5 

some    wet  covered 

1 

1 

with  dust^and  some 
pebbles  on  surface. 

448 

§8i7 

40i8 

led  Eoads. 

very  solid,with  large 
pebbles  on  level  of 

W6t  SUrfilCG  *             » 

1 
541 

1 

46.8 

1 
49.1 

1 

D 

3 

solid,  slightly  trav- 
elled, and  soft  mud. 

1 

848 

1 
30.1 

1 
Tl 

solid,  with  ruts  and 
mud 

1 
28.5 

1 
24.6 

1 
25.2 

W  is  width  of  tire.    The  other  notations  are  given  in  Art.  276. 


DRAUGHT   OF   VEHICLES. 


361 


FOE  DIFFERENT  SOILS  AND  VEHICLES. 


"  Charrettes  de  roulage." 
W=0.82ft  to  0.89ft. 

Carts. 

Diligence  of  the 
imperial  and  gen- 
eral   coach   estab- 

Wagon on 
springs  with 
seats. 

r1=0104ft. 

W=0.33ft.  to  0.39ft. 

lishment. 

W=.229  to  .262ft. 
r  —  0  087ft. 

r'=1.476ft 

r'=1.80ft. 

r\__L  1  ^ 

W=0.33ft.  to  0.39ft. 

r"=2.480ft. 

r"'=2.79ft. 

r'=2  62 

r-'=8.28 

r,=0.104ft. 

r'=2.29ft. 

r'+r"=3.936ft 

r'+r"=459ft. 

/r^.00682 

//•1=O06S2 

r'+r"=3.77ft. 

/•'  +r"  —  ?.77fti 

/r1=.00682 

yj-1=.00682 

^=0,00682 

yrj=.00574 

1 

27.2 

1 
8L7 

1 
86.8 

1 
45.4 

walk  &  trot      ^ 

walk  &  trot 

1 

26.4 

105 

12^8 

sr 

m 

walk  &  trot      ^ 

walk  &  trot 

ioi 

1 
&f 

1 
10.4 

i 

11.9 

i 

149 

1 
walk  &  trot      "g-g 

walk  &  trot 

1 
8.6 

1 

1 

1 

1 

1 

1 

"as 

ill 

walk  &  trot      ~g" 

walk  &  trot 

~T 

i 

i 

i 

I 

1 

143 

f6.7 

IlT 

23.8 

iar 

1 
7.9 

1 

1 
105 

1 

ial 

walk  &  trot       »y 

walk  &  trot 

i 

6.9 

i 

'walk              JJ-Q 

walk 

1 
49 

1 

1 

1 

i 

1 

1 

49^9 

"58 

6O2 

8^8 

.  trot                40.9 

trot 

4T8 

1 

1 

fast  trot         89j 

fast  trot 

406 

1 

1 

walk              387 

walk 

848 

1 

1 

1 

1 

1 

1 

35^2 

IT 

47.0 

58i6 

trot                2678 
1 

trot 

2L2 

1 

fast  trot         248 

fast  trot 

24.6 

1 

walk 

1 

40.8 

4T8 

1 

i 

1 

1 

i 

trot                26~5 

trot 

1 

4278 

49^8 

5^9 

7T 

fast  trot 

fast  trot 

27 

1 

22.6 

22.8 

:walk 

walk 

1 

26.1 

204 

1 

1 

1 

1 

1 

trot 

1 

21.7 

22 

2L2 

81J 

86^2 

4^2 

1 

1 

fast  trot          2^- 

fast  trot 

20.3 

1 

1 

"walk              -gj- 

walk 

2T5 

1 

1 

1 

1 

1 

1 

.  trot           ias 

trot 

185' 

22^2 

25.8 

29^5 

86.9 

i 

1 

fast  trot         iTl 

fast  trot 

17.2 

362 


DRAUGHT   OF  VEHICLES. 


Artillery  carriages 
and  carts. 

Artillery  wagons. 

"  Comtois  "  wagons. 

Designation  of  the  route 
passed  over  by  the 
carriage. 

I 

W=0.82ft.  to  0.89ft. 
r,=0.124ft. 
r=2.565ft. 
r  +r"=5.130ft. 

W=0.229ft.  to  0.246ft. 
r1=0.124ft. 
*-'=1.885ft. 
r"'=2.558ft. 
r'+r"=4.4426ft. 

W=0.196ft.  to  .229ft. 
r,=0.087ft. 
f=2.049ft. 
r"=:2.377ft, 
r'+r"=4.426ft. 

1 

/r^.00806 

/r^.00806 

/r1==.00574 

( 

with    detritus     and 

1 

1 

1 

thick  mud 

241 

20T8 

2T3 

FH 
13 

<3>   ' 

much  worn,deep  ruts 
from  .196ft.to.  262ft. 

1 

1 

1 

3 

2 

thick  mud. 

18.4 

15.9 

16^2 

<s 

3 

in   bad   order,  deep 

ruts  from  0.32ft.  to 

1 

1 

1 

0.39ft.  thick  mud, 
bottom  hard   and 
uneven  

16.5 

14.8 

1474 

Pav 

fr< 

ed    with    sandstone 
)m  Sierck 

1 

80.9 

1 
70 

1 
75.5 

1 

1 

1 

i 

with  usual  dryness 

75.7 

64.6 

69.2 

422 

1 

Is 

Ditto. 

74.7 

11 

II 

*rt 

I 

In  good  order,  wet 
and  covered  with 
mud 

1 
58.1 

1 
508 

1 
52.9 

Flo( 

>ring  of  bridge  with 

1 

1 

1 

jo 

ists  

54.1 

46TS 

49.1 

287.  General  conclusions. — From  a  general  inspection 
of  all  experiments,  we  derive  the  following  practical 
laws: 

1st.  The  resistance  opposed  to  the  rolling  of  wagons, 
by  solid  metalled  roads  or  by  pavements,  and  referred  to 
the  axis  of  the  axle,  in  a  direction  parallel  to  the  ground, 


DKAUGHT   OF   VEHICLES. 


363 


"  Charrettes  de  roulage." 
W=0.32ft.  to  0.39ft. 

Carts. 

Diligence  of  the 
imperial  and  gen- 
eral   coach   estab- 

"Wagon on 
springs  with 

seats. 

/•!=:0.104ft. 

W=0.33ft.  to  0.39ft. 

lishment. 

W=.229  to  .262ft. 

r1=0.104ft. 

W=0.83ft.  to  0.39ft. 

rj=0.087ft. 
r'=1.4Sft. 

r'=1.476ft 

r'=1.80ft. 

p"=2.480ft. 

r"=2.79ft. 

'  r'=2.62 

r'  —  3  gg 

r=.893 

ft  +  f"  —  3  936ft 

r'+r"=4.59ft. 

//•,=a.0682 

1/>'1=0.06S2 

r>  _j_^,"_lg  77ft 

/r^.00682 

/r^.00682 

/r1=0.00682 

y^jszO.00688 

fwalk 

walk 

1 

17.9 

8  1 

1 

1 

1 

1 

1 

_i^ 

18J 

2T8 

2lL9 

31.1 

isTs 

trot 

j  smart  trot     —  - 

smart  trot 

i 

I                       14.9 

15 

f                         1 

i 

walk              jg-7 

walk 

las 

1 

1 

1 

1 

trot               JL 

trot 

i 

14.8 

16.7 

IsT 

28.8 

12.4 

laB 

quick  trot      _L 

quick  trot 

i 

11.8 

.1.8 

1 

1 

i 

1 

(j_ 
Wftlk               12~.2 

walk 

1 
12JJ 

12.7 

14.9 

17 

2T2 

1 

1 

trot                ^ 

trot 

9i9 

walk              -1. 

walk 

1 
64.2 

1 

1 

1 

1 

.  trot                 JL 

;rot 

1 

64.7 

75.5 

86.3 

107.9 

42 

43" 

1 

1 

fast  trot         gg-g 

fast  trot 

"37 

1 

I 

walk              ^Q 

walk 

59 

1 

1 

1 

1 

trot 

trot 

1 

59.6 

69.5 

79.9 

99.9 

88.1 

"89" 

fast  trot          — 

fast  trot 

1 

32.  i 

sag  i 

Walk              5771 

walk 

i 

59 

trot 

1 

40.9 

trot 

4L8 

fast  trot 

fast  trot 

1 

85.8 

86T5 

walk              — 

walk 

1 

44 

45.1 

1 

1 

1 

1 

1 

trot 

1 

00    K 

46 

53.5 

74.4 

76.5 

1 

oo.O 
1 

fast  trot          2^2 

fast  trot 

2^8 

1 

1 

1 

1 

1 

1 

4^8 

w 

n 

walk  &  trot  ^ 

walk  &  trot 

a3| 

is  sensibly  proportional  to  the  pressure  or  total  weight 
of  the  vehicle,  and  inversely  proportional  to  the  diameter 
of  the  wheels. 

2d.  Upon  paved  ov  metalled  causeways,  the  resistance 
is  very  nearly  independent  of  the  width  of  the  tires. 

3d.  Upon  compressible  bottoms,  such  as  earths,  sands, 


364  DRAUGHT   OF   VEHICLES. 

gravel,  etc.,  the  resistance  decreases  with  the  increase  of 
width  of  tire. 

4th.  Upon  soft  earths,  such  as  loam,  sand,  earth-drift- 
ways, etc.,  the  resistance  is  independent  of  the  velocity. 

oth.  Upon  metalled  roads  and  upon  pavements,  the 
resistance  increases  with  the  velocity.  The  increase  is  sp 
much  less,  as  the  wagon  is  better  hung,  and  the  road  more 
smooth. 

6th.  The  inclination  of  the  draught  should  approach 
the  horizontal  for  all  roads  and  for  common  wagons,  as 
far  as  the  construction  will  admit.  Let  us  bear  in  mind 
that  these  simple  laws  are  not  strictly  mathematical,  but 
merely  approximate,  which,  for  the  most  common  cases 
of  practice,  and  for  the  usual  dimensions  of  carriages, 
will  represent  the  results  of  experiment,  with  sufficient 
exactness,  and  very  nearly  equal  to  what  may  be  deduced 
from  the  direct  experiment.  It  is  in  this  sense  only  that 
I  have  proposed  and  applied  them. 

288.  Consequences  relative  to  the  construction  of  vehi- 
cles.— From  what  precedes,  it  follows  that  the  transporta- 
tion business  has  for  its  interest  to  use  for  vehicles,  wheels 
of  the  greatest  diameter  comporting  with  their  construc- 
tion and  destination.  Carts  being  more  readily  adapted 
than  other  two- wheeled  vehicles  to  the  use  ot  great  diam- 
eters, afford  in  this  regard  a  marked  advantage.  But,  on 
the  other  hand,  if  the  roads  are  in  bad  condition,,  pro- 
ducing jolts,  the  shaft  horse  being  knocked  about  by  the 
thills,  becomes  fatigued,  is  soon  ruined  if  he  is  spirited,  or 
if  lazy  will  leave  the  draughting  to  be  done  by  the  other 
horses. 

Now,  by  bringing  the  axle  of  the  hind  wheels  nearer 
to  the  forward  wheels,  and  thus  placing  them  more  under 
the  load,  the  proportion  of  the  load  borne  by  the  hind 
wheels  will  be  more  considerable,  and  so  the  draught  will 
be  diminished.  "We  may  then  considerably  reduce  the 


DRAUGHT   OF   VEHICLES.  365 

draught  of  the  small  wheels,  which  will  thus  be  relieved, 
and  the  chariot  is  nearly  transformed  to  the  cart.  Still, 
we  must  place  in  the  front  a  sufficient  preponderance  of 
the  load,  so  that  in  ascending,  the  box  may  run  no  risk 
of  rising  and  turning  around  the  hind  axle.  This  obser- 
vation shows  that  the  weighing  of  vehicles  in  the  lump 
would  be  fallacious,  if  it  is  pretended  that  the  weight  is 
distributed  in  equal  parts  upon  each  wheel.  It  is  well 
known  that  wagoners  have  for  a  long  time  appreciated 
the  necessity  of  loading  the  hind  truck  in  a  much  greater 
proportion  than  the  fore  truck.  But  we  see  that  for  a 
given  and  nearly  constant  distribution  of  the  load,  as  with 
diligences,  omnibuses,  etc.,  there  is  an  advantage  in 
bringing  the  hind  axle  as  far  as  possible  under  the  vehicle, 
and  this  explains  why,  all  else  being  equal,  short  vehicles 
require  so  much  less  draught  than  long  ones.* 

289.  Destructive  effects  produced   lyy  vehicles  upon 
roads. — The  destructive  influence  of  vehicles  upon  roads 
has  for  a  long  time  called  for  the  attention  of  governments 
and  of  engineers ;  but  whatever  may  have  been  the  im- 
portance of  this  question,  for  the  interest  of  the  state  and 
trade,  there  has  been,  to  the  present,  but  little  time  de- 
voted to  a  profound  study  of  facts,  in  place  of  which  we 
have  theoretical  considerations  more  or  less  plausible,  but 
often  quite  contradictory  to  nature.     Without  entering 
into  a  discussion  which  would  take  us  beyond  our  pro- 
posed limits,  we  propose  to  examine  successively  the  con- 
sequences which  we  may  deduce  from  direct  experiments 
upon  the  draught,  as  far  as  concerns  the  preservation  or 
destruction  of  roads,  and  we  will  afterwards  publish  the 
principal  facts  which  we  have  directly  observed. 

290.  Preserving  influence  of  great  diameters  of  wheels. — 

*  For  further  details  see  "les  Experiences  sur  le  tirage  des  voitures  et  sur 
les  effets  destructeurs  qu'elles  exercent  sur  les  routes." 


366  DRAUGHT   OB'  VEHICLES. 

The  resistance  experienced  by  a  wheel  from  the  ground, 
being  evidently  a  more  or  less  immediate  measure  of  the 
efforts  of  compression  or  of  disintegration  which  it  exerts 
upon  the  ground,  we  see  at  once  that,  since  wheels  of  a 
great  diameter  occasion  less  draught  than  those  of  small 
diameter,  they  must  also  produce  less  disintegration  upon 
roads.  A  very  simple  observation  confirms  this  fact.  If 
we  take  stones  from  0.22ft-  to  0.26ft-  diameter,  and  upon  a 
road  somewhat  wet  and  soft,  place  some  before  the  small 
wheels  of  a  diligence,  and  others  before  the  great  wheels, 
we  see  the  first  are  pushed  forward  by  the  small  wheels 
penetrating  the  ground,  which  it  ploughs  and  disin- 
tegrates, while  the  second,  simply  pressed  and  borne  down 
by  the  great  wheels,  seldom  experience  a  displacement. 

This  results  evidently  from  the  fact  that  if  we  resolve 
the  effort  exerted  by  the  wheel,  upon  the  stone  at  the 
point  of  contact,  into  two  others,  the  one  vertical,  tending 
to  bury  the  body  in  the  ground,  the  other  horizontal, 
tending  to  push  it  forward,  the  second  effort,  which  pro- 
duces the  tearing  away  of  the  road,  is  evidently  much 
greater  proportionally  for  the  small  wheels  than  for  the 
great. 

From  this  simple  observation  we  infer,  as  I  had  already 
done  in  1838,  that  the  effects  of  disintegration  produced 
~by  the  wheels  of  vehicles,  are  so  much  the  greater,  as  the 
wheels  are  smaller. 

Experiment  having  also  proved  that  the  draught  on 
solid  bottoms  increases  but  slightly  with  the  decrease  in 
breadth  of  tire,  we  may  also  infer  that  the  loads  capable 
of  producing  equal  destruction  should  not  increase  pro- 
portionably  with  the  width  of  the  felloes,  as  all  the  rules 
of  the  carriage  police  have  admitted,  and  that  the  loads 
permitted,  according  to  these  rules,  for  the  broadest 
wheels,  must  produce  more  wear  than  those  with  narrow 
wheels. 

Finally,  the  resistance  increasing  with  the  velocity,  it 


DRAUGHT   OF   VEHICLES.  367 

was  natural  to  suppose  that  wagons  going  at  a  trot  do 
more  mischief  to  roads  than  those  going  at  a  walk.  But 
suspension,  by  diminishing  the  intensity  of  the  shocks, 
may  compensate  for  the  effects  of  velocity  in  certain  pro- 
portions. 

291.  Direct  experiments  upon  the  destructive  effects 
produced  by  wagons  upon  roads. — However  rational  these 
deductions  from  experiments  upon  the  draught  of  vehicles 
may  seem,  it  was    necessary  to  verify  them   by  other 
special  experiments,  executed  upon   a  great  scale,  and 
having  in  view  a  direct  study  of  the  destructive  effects 
exerted  upon  routes  by  vehicles,  according  to  their  differ- 
ent proportions  and  the  circumstances  of  their  action. 

These  experiments  commenced  at  Metz  in  1837,  by 
order  of  the  minister  of  war,  were  continued  in  the  envi- 
rons of  Paris  in  1839,  1840,  and  1841,  by  order  of  the 
minister  of  public  works. 

To  distinguish  the  separate  influences  of  the  width  of 
felloes,  of  the  diameter  of  wheels,  and  of  the  velocity, 
upon  the  wearing  of  roads,  I  have  studied  their  respective 
effects,  and  to  establish  them,  I  made  use  of  a  direct  ab- 
stract of  the  route  by  means  of  cross  sections,  and  a 
measurement  of  the  draught  before,  during,  and  after  the 
experiments,  and  in  a  great  number  of  instances  by  the 
measure  of  the  quantity  of  materials  used  in  repairs. 

The  general  method  of  experimenting  consisted  in 
causing  the  vehicles  to  circulate  upon  a  particular  track, 
always  the  same,  and  kept,  by  sprinkling,  in  a  nearly 
equal  state  of  moisture  for  all,  until  the  same  total  weight 
had  been  transported  upon  each  track,  and  this  total  weight 
nearly  always  was  from  11  to  13,000,000  pounds,  and 
often  beyond  that. 

292.  Experiments  upon  the  influence  of  the  width  of 
tires. — All  the  rules  of  the  administration  and  the  laws 


368  DRAUGHT   OF   VEHICLES. 

proposed  by  the  carriage  police  having  admitted  that,  to 
obtain  for  all  vehicles  an  equal  action  upon  the  roads,  it 
was  necessary  to  charge  them  with  loads  proportioned  to 
their  widths  of  tire,  it  became  necessary  to  see  if  this 
basis  of  tariff  was  exact.  For  this  purpose,  three  artil- 
lery train  wagons,  having  each  wheels  about  4.75  ft.  diam- 
eter, for  the  fore  and  hind  trucks,  with  tires  of  0.196  ft., 
0.362ft.,  and  0.573  ft.,  were  loaded,  proportionably  to 
these  widths,  with  the  following  weights  respectively : 

Carriage  No.  1,  with  tires  of  0.196ft-,     5309  pounds. 
Carriage  No.  2,     «      "      «  0.362f%  10129      " 
Carriage  No.  3,     «      «      «  0.573ft-,  15417      " 

These  carriages,  thus  loaded,  were  made  to  traverse 
three  tracks,  each  984  ft.  in  length.  On  account  of  farms 
bordering  the  road,  it  happened  that  the  track  of  No.  1, 
with  narrow  tires,  was  generally  more  moist  than  that  of 
the  other  two,  and  that  consequently  this  carriage  was  in 
less  favorable  circumstances. 

Observation  has  shown  that  the  draught  upon  the 
track  of  carriage  No.  3,  with  broad  tires,  was  increased 
with  the  number  of  trips,  much  more  rapidly  than  upon 
the  other  two  tracks,  that  it  was  also  increased,  but  in  a 
much  less  ratio,  upon  the  track  of  carriage  No.  2,  with 
tires  0.362  ft.,  and  that  finally,  upon  the  track  of  carriage 
No.  1  it  remained  stationary,  and  only  varied  by  reason 
of  the  state  of  moisture  of  the  road. 

Moreover,  an  examination  of  the  state  of  the  road,  of 
the  abstract  of  the  cross  sections,  and  of  the  measurement 
of  the  draught,  all  -agree  in  showing  that  after  the  trans- 
portation of  the  same  weight  of  materials  the  carriage  No. 
3,  with  tires  of  0.573  ft,  loaded  with  15417  pounds,  car- 
riage included,  produced  much  more  injury  than  the  two 
other  carriages ;  that  the  carriage  No.  2,  with  tires  of 
0.362ft,,  loaded  with  10129  pounds,  had  produced  more 
than  carriage  No.  1,  with  tires  of  0.196ft.,  loaded  with 


DRAUGHT   OF   VEHICLES.  369 

5309  pounds,  and  that  the  last  produced  no  ruts  and  no 
apparent  wear. 

293.  Consequences  of  these  experiments. — It  seems  then 
that  we  may  conclude  from  these  experiments,  made  upon 
carriages  exactly  alike  in  all  respects,  saving  in  the  width 
of  tires  and  magnitudes  of  loads,  which  were  proportional 
to  this   width,  that  the  proportionality  of  loads  to  the 
widtJis  of  tires,  admitted  as  the  basis  of  tariffs,  was  more 
unfavorable  than  useful  to  roads. 

294.  Experiments  made  with  the  same  carriages  under 
equal  loads. — Two  similar  experiments  were  made  upon 
the  same  carriages,  loaded  with  an  equal  weight  of  12228 
pounds,  carriage  included,  and  upon   three   tracks  also 
identical  as  possible,  and  always  kept  very  moist  by 
abundant  sprinkling,  they  were  made  to  circulate  until 
they  had  transported  each,  18356625  pounds. 

The  abstract  of  profiles,  and  above  all  the  result  of  the 
experiments  in  traction,  have  shown  that,  with  equal 
weights,  upon  metalled  or  gravelled  roads,  the  wheels, 
with  tires  of  0.196  ft.,  produced  disintegration  far  more 
considerable  than  those  of  0.362ft.,  but  that  beyond  this 
last  width  there  is  very  little  advantage  in  the  interest  of 
maintenance  of  roads  to  increase  the  dimensions  of  the 
rim  of  the  wheel. 

295.  Experiments  upon  the  influence  of  the  diameter 
of  wheels,  in  their  destructive  effects  upon  roads. — Similar 
experiments  have  been  made  with  the  same  carriages 
with  wheels  of  a   common  width  of  0.362  ft.,  but  with 
diameters  varying  from  2.859  ft.  to  4.766  ft.  and  6.655  ft., 
and  which  were  loaded  with  the  same  weight,  equal  to 
10870  pounds.     The  tracks  passed  over  by  these  carriages 
were  656  ft.  in  length,  and  they  were  sprinkled  during  the 
last  part  of  the  period  of  the  experiment.     An  examina- 

24 


370  DRAUGHT   OF  VEHICLES. 

tion  of  the  road,  of  the  abstract  of  the  profiles,  and  the 
measure  of  the  intensity  of  draught,  proved  by  the  self- 
same carriages  upon  three  tracks  after  a  transportation  of 
22040562  pounds  upon  each  of  them,  has  shown  that  the 
track  passed  over  by  the  carriage  with  the  small  wheels, 
2.859  ft.  in  diameter,  was  much  the  most  worn,  and  that 
of  the  carriage  with  great  wheels  of  6.655  ft.  diameter, 
was  nearly  intact ;  which  evidently  proves  the  considera- 
ble advantage  afforded  by  great  wheels  for  the  preserva- 
tion of  roads. 

296.  Influence  of  velocity  upon  the  destructive  effects. — 
It  was  proposed  to  compare  the  injury  produced  upon 
roads  by  carriages  on  springs,  going  at  a  trot,  with  those 
occasioned  by  wagons  unhung,  going  at  a  walk.  For  this 
purpose,  we  used  two  chariots  exactly  similar,  one  of 
which  was  suspended,  and  the  other,  by  the  wedging  of 
the  springs,  was  transformed  into  a  chariot  unhung.  The 
load  of  these  two  vehicles  was  at  first  fixed  at  13232 
pounds,  vehicle  included,  then  at  11027  pounds,  when  the 
road  was  in  bad  condition. 

The  chariot  suspended  was  carried  at  a  trot  of  10.5  ft. 
to  11.81  ft.  in  1",  or  from  7.15  to  8.05  miles  per  hour,  and 
the  chariot  not  suspended  at  a  gait  of  3.28  ft.  to  3.937  ft. 
per  1",  or  from  2.23  to  2.68  miles  per  hour. 

An  examination  of  the  road,  the  abstract  of  profiles,  and 
the  measurement  of  the  draught,  has  shown  that  the  dis- 
integration, as  well  as  the  increase  of  draught  upon  the 
two  tracks,  was  sensibly  the  same  after  the  transportation 
of  about  10253000  pounds  upon  each  of  them.  These 
results  have  completely  confirmed  the  experiments  which 
had  previously  been  made  at  Metz,  and  prove  that,  in 
considering  only  the  preservation  of  roads,  the  law  should 
not  impose  upon  spring  carriages  going  at  a  trot  more 
restricted  limits  than  upon  wagons  without  springs  going 
at  a  walk. 


DRAUGHT    OF    VEHICLES.  371 

297.  Comparative  experiments  upon  the  wear  pro- 
duced J>y  carriages  (comtois),  carts,  and  wagons,  without 
springs. — It  was  admitted,  for  a  long  time,  that  one  horse 
wagons,  and  voitures  "  comtois"  with  narrow  tires,  were 
more  destructive  to  roads,  than  the  large  wagons  and 
carts  with  broad  tires,  drawn  by  several  horses,  and  in 
1837  these  vehicles,  so  light,  so  useful,  and  by  means  of 
which  the  power  of  horses  is  so  well  utilized,  came  near 
being  abolished.  The  results  of  our  first  experiments 
upon  the  influence  of  the  widths  of  tires,  have  doubtless 
sufficed  to  show  how  erroneous  these  impressions  were, 
but  it  seemed  none  the  less  useful  to  institute  a  series  of 
direct  experiments,  upon  business  wagons,  loaded  in  their 
usual  proportions. 

For  this  purpose,  I  procured  four  wagons,  with  tires 
0.196ft.  broad,  having  fore  wheels  of  3. 64ft.,  and  hind 
wheels  4.46ft.,  while  the  true  Franche-Comtois  wagons 
have  wheels  with  diameters  respectively  4.26ft.  arid 
4.75  ft.,  and  are  in  the  most  favorable  conditions.  Each 
of  these  wagons,  when  empty,  weighed  1378  Ibs.,  and 
when  loaded  3971  Ibs. 

A  cart  with  wheels  6ft.  diameter  and  0.54ft.  breadth 
of  tire,  weighing,  empty,  2260  pounds,  and  when  loaded 
11044  pounds,  and  a  wagon  with  wheels  3.313  ft,  and 
5.675  ft.  in  diameter,  with  0.54  ft.  breadth  of  tire,  weigh- 
ing, when  empty,  7000  pounds,  and  when  loaded  17496 
pounds,  were,  along  with  the  chariots  (comtois),  put  on 
trial,  upon  three  tracks  492  ft.  in  length,  all  in  the  same 
condition,  and  kept  constantly  wet. 

By  means  of  observations  made  in  the  previous  experi- 
ments we  have  connected  with  them  a  measurement  of 
the  quantity  of  material  required  for  repairs.  From  all 
these  means  united  it  resulted  : 

1st.  That  the  (comtois)  wagons,  after  a  transportation 
of  about  15438360  pounds,  upon  a  tivsk  always  wet,  pro- 
duced less  wear  than  the  wagon  and  f  «  cart. 


372  DRAUGHT   OF   VEHICLES. 

2d.  That  in  the  same  circumstances,  the  wagon  with 
four  wheels  produced  less  wear  than  the  cart. 

These  results  prove,  incontestably,  the  advantage  of  a 
division  of  the  load  upon  wagons  with  narrow  tires,  and 
they  demonstrated  that  the  transportation  of  heavy  goods 
should  be  made  in  four-wheeled  wagons. 

All  the  consequences  just  published  do,  then,  justify 
and  establish  those  derived  solely  from  the  experiments 
upon  traction. 

298.  Experiments  to  determine  the  loads  of  equal  wear 
and  tear. — The  researches,  whose  results  we  have  so  mi- 
nutely recorded ,  having  proved  that  the  bases,  until  then 
admitted  in  the  laws  and  regulations  of  the  carriage 
police,  were  incorrect  and  incomplete,  it  becomes  us  to 
investigate  the  new  ratios  to  be  established  between  the 
dimensions  of  wheels,  as  to  their  diameter,  width,  and 
loads,  so  that  all  the  vehicles  employed  in  trade  may 
produce  nearly  the  same  wear  upon  the  roads. 

New  experiments  were  made  at  Courbevoie  in  1841, 
and  their  results  served  as  bases  for  the  draft  of  a  law 
presented  to  the  Chambers  in  1842,  and  especially  for  the 
report  of  the  commission  of  the  chamber  of  deputies,  who 
had  studied  the  question  conscientiously. 

This  is  no  place  to  enter  into  a  detailed  study  of  these 
researches,  which  belong  more  properly  to  legislation  and 
public  administration  than  to  industrial  mechanics,  and  I 
shall  limit  myself  with  a  reference  to  a  publication  made 
by  me  in  1842,  under  the  title  of  "  Experiences  sur  le 
tirage  des  voitures  et  sur  le&  effets  destructeurs  qu'elles 
exercent  sur  les  routes." 


KESISTA1STCE  OF  FLUIDS. 

299.  The  resistance  of  fluids. — When  a  body  moves 
in  a  fluid  mass,  it  necessarily  displaces  the  molecules 
of  the  latter,  impressing  upon  them  velocities  in  a 
certain  ratio  to  its  own,  and  we  easily  conceive  that  the 
inertia  of  the  molecules,  thus  set  in  action,  develops  a 
resistance  which  increases  with  the  velocity  of  the  body. 
Similar  effects  are  produced  when  a  body  at  rest  or  in 
motion  is  shocked  by  a  fluid. 

The  manner  in  which  the  fluid  particles  are  divided  at 
their  meeting  with  the  body,  depends -much  upon  the 
form  and  proportions  of  the  latter,  and  we  see  that  the 
resistance  in  question  must  vary  notably  with  these  cir- 
cumstances. 

This  important  question  of  physical  mechanics  has  for 
a  long  time  engaged  the  attention  of  philosophers,  and  we 
find  in  the  introduction  to  the  industrial  mechanics  of  M. 
Poncelet,  a  complete  analysis  of  all  the  ancient  and 
modern  researches  upon  this  matter.  I  only  propose,  in 
these  lectures,  to  call  attention  to  the  most  important 
cases  in  practice. 

300.  Theoretic  considerations. — When  a  body  of  any 
form  mnpq  (Fig.  87)  moves 
in  a  stream  in  a  direction  xy, 
if  we  project  it  upon  a  plane 
perpendicular  to  the  direction 
of  motion,  it  is  easy  to  see 


374  RESISTANCE   OF   FLUIDS. 

that  ill  moving  an  elementary  space  s=inm',  the  body 
will  displace  a  volume  of  liquid  which  will  be  represented 
by  As,  obtained  from  multiplying  the  projection  of  the 
body  upon  the  plane  perpendicular  to  the  direction  of  its 
motion,  by  the  path  described.  In  fact,  in  its  two  suc- 
cessive positions  the  body  occupies  in  the  fluid,  the  same 
volume,  and  in  each  of  them  there  is  a  part  of  this  vol- 
ume which  has  not  changed  its  position,  which  corres- 
ponds to  mop'r,  so  that  the  anterior  volume  on'm'q'rm  is 
necessarily  equal  to  the  volume  onpqrp ' . 

It  is  no  less  evident  that  each  of  these  volumes  is  also 
equal  to  the  volume  mn'q'q,  made  by  the  greatest  cross 
section  of  the  body,  or  by  the  area  of  its  projection  A ; 
for  these  three  elementary  volumes  may  be  regarded  as 
composed  of  an  infinity  of  small  prisms  of  the  same  base, 
height,  and  number,  whose  edges  are  parallel  to  the  direc- 
tion of  motion,  and  which  only  differ  in  their  respective 
positions. 

Thus,  when  the  body  describes  in  relation  to  the  fluid, 
or  when  the  fluid  describes  in  relation  to  the  body,  an 
elementary  space  s,  the  volume  of  the  deviating  fluid, 
which  passes  from  the  front  to  the  rear  of  the  body,  is  ex- 
pressed by  q=As,  and  its  mass  is 


calling  d  the  density  or  weight  ot  a  cubic  foot  of  the 
fluid  ;  this  deviating  mass  effects  its  relative  displacement 
with  a  velocity  depending  essentially  upon  that  of  the 
body  in  relation  to  the  fluid,  in  the  case  when  it  is  the 
body  that  moves,  and  which  it  is  natural  to  suppose  is 
proportional  to  the  velocity  of  the  body.  It  will  be  the 
same  for  the  vis  viva  imparted  to  the  deviating  fluid  ;  so 
that  in  the  case  of  a  fluid  at  rest,  in  which  moves  a  body 
impressed  with  a  velocity  Y,  the  vis  viva  imparted  to  the 


RESISTANCE   OF  FLUIDS.  375 

displaced  fluid,  for  an  elementary  motion  of  the  body  will 
be  proportional  to 


and  if  we  call  Ic  the  unknown  ratio,  (to  be  determined  by 
experiment)  of  the  vis  viva  F  really  impressed  upon  the 
fluid,  to  the  above  expression,  we  shall  have 


On  the  other  hand,  if  we  call  R  the  total  resistance 
which  the  inertia  of  the  fluid  molecules  opposes  to  their 
displacement,  the  work  of  this  resistance  for  the  elemen- 
tary displacements  will  be  Rs,  and  must,  according  to  the 
general  principle  of  vis  viva,  be  equal  to  one-half  of  the 
vis  viva  imparted  to  the  displaced  fluid.  We  should  then 
have  the  relation 


whence 

MAY3 


In  the  case  where  the  fluid  displaced  by  the  body  is 
moving  with  a  velocity  of  its  own,  if  the  body  moves  in 
an  opposite  direction  to  the  motion  of  the  fluid,  the  rela- 
tive velocity  with  which  the  molecules  are  met  and  dis- 
placed is  Y+v  ;  and  when  the  two  velocities  Y  and  v  are 
in  the  same  direction,  the  relative  velocity  is  Y—  v  ;  simi- 
lar reasoning  to  that  already  used  will  give  us  then,  for 
the  case  in  which  the  body  moves  in  an  opposite  direction 
to  the  fluid 


and  for  that  in  which  they  move  in  the  same  direction 

. 


376  RESISTANCE  OF  FLUIDS. 

301.  Work  developed  per  second  ~by  the  resistance  of  a 
medium.  —  When  all  the  circumstances  of  motion  remain 
the  same,  and  the  phenomena  occur  constantly  in  the 
same  manner,  the  work  developed  in  each  second  by  the 
resistance  of  the  medium  opposed  to  the  motion  of  the 
body  is,  in  the  case  of  a  fluid  at  rest, 


and  in  the  case  of  a  fluid  in  motion 


, 

which  shows  that,  in  the  first  case,  the  work  of  the  resist- 
ance increases  as  the  cube  of  the  velocity. 

302.  Equivalent  expressions  of  the  resistance.  —  In  the 
preceding  expression  of  the  resistance,  applied  to  a  liquid 
whose  density  d  is  constant,  and  in  places  where  the  value 

of  2^=64,3634,  we  may  place  0-=^,  and  it  then  takes  the 

form 

K=K.AV2, 
or 


in  which  it  is  frequently  used. 

Some  authors,  and  in  particular  Dubuat,  calling  H  the 
height  due  to  the  relative  velocity  V  or  V±  v,  and  put- 

ya  (V+vf 

ting,  consequently,  H=  —  ,  or  H=v—  -—  ,  and  K'= 
2<7  2<^ 

write  this  formula  under  the  form 


It  is  evident  that  the  three  formulas  are  equivalent, 
and  I  have  pointed  out  the  two  last,  which  less  convey 


RESISTANCE   OF   FLUIDS.  377 

the  idea  of  the  law  of  resistance,  only  to  facilitate  the  un- 
derstanding of  other  authors. 

303.  Case  of  a  ~body  at  rest  in  a  fluid  in  motion. — The 
body  being  at  rest  in  the   fluid,  if   we   suppose   them 
to   be  impressed  with  a  common  motion  of  translation, 
whose  velocity  is  precisely  equal  and  opposite  to  the  actual 
velocity  of  the  fluid  ;  this  would  give  us  the  fluid  at  rest, 
and  the  case  would  revert  to  the  preceding ;  therefore, 
the  expression  for  the  resistance  should  be  the  same. 

The  consideration  of  the  physical  phenomena  presented 
by  the  displacement  of  the  fluid  molecules  situated  in 
front  of  the  body,  and  the  return  of  those  flowing  to  the 
rear  to  fill  up  the  void  formed  by  its  passage,  induced 
Dubuat  to  infer  that  the  resistance  experienced  by  a  body 
moving  in  a  fluid  at  rest,  was  not  the  same  as  the  effort 
exerted  upon  a  body  at  rest,  by  a  fluid  in  motion,  all  else 
being  equal.  This  would  be  contrary  to  the  preceding 
results,  but  there  is  further  need  of  its  confirmation  by 
experiments ;  it  seems,  however,  to  accord  with  those  of 
M.  Thibault  upon  the  resistance  of  air,  which  we  shall 
notice  hereafter.  However,  the  difference,  if  it  exists, 
must,  in  most  cases,  be  so  small,  that  it  may  be  neglected. 

304.  Experiments  upon  the  resistance  of  water  to  the 
motimi  of  variously  formed  bodies. — Though  these  experi- 
ments may  be  of  small  importance  in  an  industrial  point 
of  view,  which  is  the  main  object  of  this  work,  I  will 
report  those  which  were  made  at  Metz,  in  1836  and  1837, 
principally  on  account  of  the  methods  of  observation  em- 
ployed. 

The  bodies  subjected  to  these  experiments  were  : 

1st.  Thin  iron  plates  of  different  sizes,  which  were 

made  to  ascend  from  the  bottom  to  the  surface  of  the 

water  by  the  action  of  a  counter  weight. 

2d.  Solid   or   hollow   brass   spheres,  with   diameters 


378  RESISTANCE   OF   FLUIDS. 

ranging  from  0.341ft,  0.387ft.,  0.422ft,,  0.485ft,  to 
0.530  ft 

3d.  Tin  plate  cylinders,  with  altitudes  equal  to  their 
diameters,  which  were  0.324ft.,  0.656ft,  and  0.984ft 

4th.  Cones  terminating  upon  cylinders,  with  the  same 
diameter  and  height  as  the  preceding,  and  whose  angles 
at  the  summit  varied  as  follows  : 

Half  angle  at  the  summit,  64°.48/;  46°.50/;  26P.01/  . 
18°.49';  14°.19'.48". 

5th.  Cylinders  of  the  same  dimensions  with  the  pre- 
ceding, and  terminated  in  front  by  hemispheres. 

305.  Mode  of  observation. — The  experiments  were 
made  upon  the  Moselle,  in  front  of  the  dam  at  Pucelles, 
in  a  place  where  the  water  was,  at  least  at  its  surface, 
nearly  without  a  current,  and  had  a  depth  of  16.4  ft  It 
was  the  most  suitable  place  that  could  be  found  in  the 
neighborhood. 

The  vertical  motion  of  the  body  was  made  in  the 
descent,  by  its  own  weight,  with  an  occasional  ballast  to 
increase  its  velocity,  and  in  the  ascent  by  means  of  a 
counter  weight.  In  all  cases,  the  law  of  the  motion  was 
observed  and  determined  by  means  of  a  chronometric 
apparatus  with  a  style,  similar  to  those  used  in  the  exper- 
iments on  friction. 

In  the  first  experiments  it  was  at  once  apparent  that 
the  resistance  of  the  water  increased  so  rapidly  with  the 
velocity,  that  the  motion  very  readily  became  uniform. 
Then,  knowing,  in  each  case,  the  velocity  and  the  motive 
weight,  and  keeping  account  of  the  passive  resistances,  it 
was  easy  to  calculate  the  value  of  the  corresponding  re- 
sistance of  the  fluid,  and  to  investigate  its  law. 

A  graphic  representation  of  the  results,  taking  the 
resistances  for  abscissae,  and  the  squares  of  the  velocities 
for  ordinates,  has  shown  in  each  case,  as  in  the  preceding, 
that  the  resistance  is  composed  of  two  terms,  the  one 


RESISTANCE   OF   FLUIDS.  379 

independent  of  the  velocity  and  simply  proportional  to 
the  wetted  surface,  the  other  proportional  to  the  square 
of  the  velocity  ;  but  here  the  first  terra  is  so  small,  that  it 
may  be  neglected  .in  relation  to  the  second,  as  soon  as  the 
velocity  has  become  3.28  ft.  per  second. 

According  to  this,  the  resistance  opposed  by  the  water 
to  bodies,  as  proved  by  these  experiments,  would  be  rep- 
resented by  the  formula 


A  being  the  projection  of  a  body  upon  a  plane  perpen- 
dicular to  the  direction  of  motion. 

The  values  of  K  derived  from  experiment,  are  entered 
in  the  following  table  : 

Values  of  the  coefficient  K  of  the  foi^mula  K=KAVa. 


Bodies  used. 

Values  of  I£. 

Thin  plates  (rising  vertically  upwards)  

2.724 

Spheres 

0.41969 

Ric'ht  cylinders  with  height  equal  to  diameter 

17715 

Cylinders  with  same  proper-   f  0.94  to  1  ")   norrest)0n(i    ]          64°  48' 

c^nes^vlSse^eightsar^to  J  405  to  1  1    inS  anSles     I         26°    1' 
the  radii  of  their  bases  in      5.92  to  1                                      18°  49' 
the  ratio  of  [7.66  to  IJ             ••*      j          14°  19'  48" 

1.8944 
1.0276 
0.90868 
0.84802 
077449 

Cylinders  of  the  same  proportion  terminated  by  spheres 

077487 

306.  Observations  upon  these  results. — The  value  of 
the  coefficient  K  found  in  the  experiments  for  thin  plates 
is  considerable,  and  nearly  double  that  found  by  Dubuat 
in  causing  a  vertical  plane  to  move  in  a  horizontal  direc- 
tion, thus  producing  a  displacement  of  water  entirely 
different  from  that  in  our  experiments,  and  occasioning 
fcthe  difference  of  results. 

It  is  remarkable  that  of  all  the  bodies  used,  the  spheres 
offered  the  least  resistance,  and  that  cylinders,  terminated 
by  half  spheres,  have  experienced  less  than  those  with 
acute  cones. 


380 


RESISTANCE   OF  FLUIDS. 


This  result  shows,  that  in  regard  to  the  resistance  of  a 
medium,  the  spherical  form  for  projectiles,  and  the  semi- 
circular for  piers  of  bridges,  are  the  most  favorable. 

307.  Influence  of  the  acuteness  of  the  angles  of  cones 
upon  the  resistance. — In  comparing  the  values  of  the  half 
angles  at  the  summit  of  the  cones,  expressed  in  fractions 
of  the  semi-circumference,  with  the  values  of  the  resist- 
ance, we  see  that  the  coefficient  K  of  this  resistance  in- 
creases proportionally  with  these  angles ;  starting  from  a 
certain  value  answering  to  the  angle  zero.  It  may  be 
given  by  the  formula 

K=0.59005 +2.29980, 

a  being  the  half  of  the  angle  at  the  summit  in  terms  of  a 
fraction  of  the  semi-circumference.  The  comparison  of 
the  values  of  K  given  by  this  formula,  with  those  deduced 
directly  from  the  experiment,  is  established  in  the  follow- 
ing table : 

Comparison  between  the  values  of  the  coefficient  K  as  de- 
duced by  formulas  and  by  experiments. 


Half  angles  at  the  summit 

Values  of  the  coefficient  K  deduced. 

* 

semi-circumference. 

From  the  formula. 

From  experiment. 

0.500 

1.739 

1.771 

0.362 

1.422 

1.394 

0.262 

1.192 

1.027 

0.145 

0.923 

0.908 

0.105 

0.831 

0.843 

0.080 

0.774 

0.774 

"We  see  that,  with  the  exception  of  the  case  relative  to 
the  cone  whose  half-angle  at  the  summit  was  measured 
by  an  arc  equal  to  0.262  of  the  semi-circumference",  the 


RESISTANCE   OF   FLUIDS.  381 

results,  including  even  that  pertaining,  to  the  plane  base 
of  the  cylinder,  are  quite  correctly  represented  by  the 
formula,  and  that  we  may  use  it  for  intermediate  cases, 
which  have  not  been  experimented  upon. 

308.  Experiments  upon  the  resistance  of  water  to  the 
motion  of  projectiles. — Without  entering  into  details,  for 
which  this  is  not  the  proper  place,  I  must  say  something 
about  the  remarkable  results  of  experiments  made  by  me 
in  common  with  MM.  Piobert  and  Didion,  at  Metz,  in 
1836,  upon  the  penetration  of  projectiles  in  water. 

These  experiments  were  made  at  the  basin  which  had 
served  for  the  beautiful  hydraulic  researches  of  MM.  Pon- 
celot  and  Lesbros,  in  firing  horizontally  beneath  the  sur- 
face of  the  water  projectiles  which  penetrate  the  water 
after  having  traversed  an  orifice  formed  by  spruce  scant- 
ling. A  horizontal  flooring,  placed  at  the  bottom  of  the 
basin,  and  marked  with  strips,  received  the  projectiles, 
which  always  reached  it  with  a  very  small  velocity. 

"We  found,  with  this  arrangement,  the  resistance  offered 
to  solid  balls  with  diameters  of  0.354  ft.,  0.328  ft.,  0.530  ft, 
and  0.72ft.,  and  to  shells  of  the  same  diameters,  having 
different  thicknesses  and  weights,  the  initial  velocities  of 
the  projectiles  varying  from  229ft.  to  1640ft.  in  V '. 

From  the  general  view  of  all  the  experiments  made, 
the  results  of  which  are  published  in  ]STo.  VII.  of  the 
"  Memorial  de  1'artillerie,"  we  conclude  that  the  resistance 
of  water  to  the  motion  of  these  projectiles  may  be  repre- 
sented by  the  formula 

K=0.453  AV21b% 
while  the  experiments  above  cited  (No.  305)  gave  us 

K=0.4197AValbs- 
On  the  other  hand,  the  ancient  experiments  due  to 


382  RESISTANCE   OF   FLUIDS. 

Newton,  and  made  by  observing  the  time  of  the  fall  of 
spheres  in  water  lead  to  the  value 

R=0.46498AV21bs, 

and  those  which  Dubuat  made  in  causing  spheres  placed 
at  the  end  of  the  arm  of  a  horse-gin  to  pass  in  a  circle 
through  the  water,  furnish  the  formula 

K=0.419TAV21bs- 

An  inspection  of  all  these  researches,  made  by  pro- 
cesses so  different,  and  within  limits  so  extended,  enables 
us  to  conclude  that,  in  liquids,  the  law  of  the  proportion- 
ality of  the  resistance  to  the  square  of  the  velocity,  is  ap- 
plicable to  spheres,  even  when  moving  with  the  highest 
velocities. 

309.  The  resistance  of  water  to  the  motion  of  floating 
bodies. — The  preceding  theoretic  considerations  apply  to 
boats  which  navigate  the  sea,  rivers,  and   canals ;  but 
their  results  are  influenced  by  different  circumstances,  of 
which  it  is  important  to  take  an  account ;  some  are  per- 
manent, others  accidental. 

310.  Influence  of  the  form   of  floating  bodies. — We 
readily  conceive  that,  when  a   floating  body  penetrates 
and  displaces  a  liquid,  in  throwing  right  and  left  the  fluid 
molecules,  the  form  of  the  prow  must  exert  a  great  influ- 
ence upon  the  facility  with  which  it  cleaves  its  way.     So 
also  the  form  of  the  stern,  in  facilitating,  more  or  less,  the 
return  of  the  liquid  to  fill  the  void  formed  by  its  passage, 
affects  the  difference  of  level  existing  between  the  bow 
and  stern,  and  consequently  the  resistance. 

It  is  clear,  from  a  mere  inspection  of  figures  111  and 
112,  that  a  boat  whose  front  forms,  in  horizontal  planes, 
are  such  that  the  fluid  filets  are  at  first  separated  by  a 


RESISTANCE   OF   FLUIDS.  383 

nearly  vertical  edge  a  (Fig.  Ill)  of  the  form  of  a  blade,  and 
then  divided  laterally  by  curves  grad- 
ually approaching  the  sides,  would        |||  a, 
experience  a  much  less  resistance, 
than  one  with  a  bow  formed  simply  of 
vertical  planes  more  or  less  inclined 
to  the  sides.     The  first  form  is  that 


to  be  given  to  fast  boats  navigating      FIG.  in.          FIG.  112. 
rivers  or  canals,  with  steam  or  horse-power. 

311.  Flat-bottomed  boats  with  raised  fronts. — Boats 
are  used  upon  rivers,  whose  fore  part  is  formed  by  the 
prolongation  of  the  bottom,  which  rises  at  an  inclination 
of  from  25°  to  30°  with  the  horizon,  and  narrowing  very 
much  in  a  horizontal  direction.  This  form  is  very  un- 
favorable for  speed,  for 
which  these  boats  are  not 
constructed,  but  they  ad-  , 


mit  of  an  easier  approach 

to  the  banks,  and  diminish  FIG.  113. 

the  violence  of  the  shocks  against  obstacles  concealed  in 

the  river.     But  one  could  scarcely  believe  that  they  are 

yet  retained  for  common  wherries  with  oars. 

In  fact,  we  readily  see  (Fig.  113)  that  the  resistance 
of  the  water  acting  horizontally,  being  decomposed  into 
two  forces,  the  one  tangential,  the  other  normal  to  the 
prow,  the  last  tends  to  raise  the  front  and  incline  the 
boat. 

This  effect  was  quite  apparent  in  the  numerous  experi- 
ments which  I  made  at  Metz,  in  1838,  upon  boats  of  this 
kind,  among  which  there  was  one  whose  length  could  be 
changed  by  the  addition  of  other  pieces.  These  experi- 
ments were  made  in  the  trench  of  the  curtain  of  Fort 
Saint-Vincent,  about  984  ft.  long  by  98.4  ft.  broad,  and 
having  a  depth  of  water  varying  from  2.62  ft.  to  3.93  ft. 
To  facilitate  various  observations  which  I  shall  speak  of 


384:  RESISTANCE   OF   FLUIDS. 

hereafter,  the  escarpment  wall  of  this  curtain  had  been 
divided  into  portions  of  32.8  ft.  in  length,  marked  by  very 
plain  vertical  lines. 

The  motion  of  the  boat  was  produced  by  the  descent 
of  a  box  loaded  with  weights,  hung  from  the  top  of  a  crane 
stationed  upon  the  parapet  of  a  neighboring  bastion,  by  a 
rope  passing  round  the  smallest  diameter  of  a  windlass 
with  two  drums.  Upon  the  largest  of  these  drums,  which 
was  8.45ft.  in  diameter,  the  smallest  being  only  1.64  ft., 
was  wound  a  towing  line  984  ft.  long,  its  end  being  fast- 
ened to  the  boat  by  means  of  a  dynamometer  with  a  style. 

An  observer  placed  in  front,  provided  with  a  watch 
indicating  tenths  of  seconds,  observed  the  instant  of  pass- 
ing the  equidistant  divisions  of  the.  escarpment,  and  thus 
determined  the  velocity,  while  he  watched  the  dyna- 
mometer. 

Finally,  to  observe  the  inclination  of  the  boat,  and 
other  circumstances  of  its  progress,  there  was  placed  at 
the  ends  of  the  stem  and  stern  two  vertical  uprights,  ter- 
minated by  small  laths  aa! ,  W  (Fig.  114,)  movable  around 

U  a  horizontal  screw. 

*•'  J  Perpendicular  to  the 

l\  Ji  direction  of  the  course 

of  the  boat,  and  in  a 
horizontal  position, 
FIG- 114  about  5.25  ft.  above 

the  surface,  there  was  firmly  fixed  a  plank,  having  at  its 
lower  edge  a  triangular  bracket  0,  whose  level  edge  was 
whitened  with  chalk,  while  the  laths  aa!  and  W  were 
blackened. 

Before  commencing  the  experiments,  the  boat  was 
gently  brought  under  the  bracket,  and  the  height  of  the 
bracket  above  the  two  points  at  the  bottom  of  the  boat, 
corresponding  with  the  laths,  was  marked.  The  two 
heights  thus  marked  were  generally  equal,  or  nearly  so, 
when  the  boat  was  at  rest. 


RESISTANCE   OF  FLUIDS.  385 

This  done,  the  boat  was  taken  to  its  starting-point,  and 
the  experiment  of  its  trip  began.  It  is  readily  seen  that 
the  small  movable  laths  coming  in  contact  with  the  fixed 
racket  and  falling,  as  they  pass,  will  preserve  the  print  of 
the  shock  against  the  bracket,  and  so  give  the  height  by 
which  each  end  was  raised  or  lowered  during  the  motion. 

By  this  arrangement,  the  inclinations  of  the  boat  for 
its  different  velocities,  could  easily  be  compared,  and  the 
amount  of  its  rise  or  fall  from  a  position  of  rest,  ascer- 
tained. The  experiments  in  question  were  made  with  a 
wherry,  with  a  deck  boat,  and  with  a  boat  with  lengthen- 
ing pieces,  having  a  breadth  of  only  1.968  ft.  at  the  bottom, 
and  2.296ft.  at  the  gunwale,  a  depth  of  2.62ft,  and  suc- 
cessive lengths  of  22.14ft.,  32.8ft.,  and  33.63ft.  The 
draft  of  water  varied  for  this  last  boat  from  0.92ft.  to 
1.38  ft.,  and  the  velocities  from  6.07  ft.  to  16.4ft.  in  1". 

The  first  fact  presented  by  these  experiments  is,  that 
the  area  of  the  greatest  submerged  section  during  the 
trip,  is  generally  superior,  or  at  least  equal,  to  that  of  the 
submerged  section  in  repose.  The  second  is,  that  the 
inclination  of  the  boat  to  the  horizon  increases,  at  first, 
rapidly  with  the  velocity,  and  then  increases  less  promptly 
with  the  velocity,  varying  with  the  length  of  the  boat 
and  the  draft  of  water ;  in  general,  it  increases  up  to  ve- 
locities of  16.4  ft.,  even  for  a  boat  whose  width  at  the  bot- 
tom is  only  T1T  of  its  length  ;  which  proves  how  ill  adapted 
this  form  is  to  fast  sailing,  and  that  even  for  wherries  it 
should  be  abandoned. 

312.  Velocity  of  waves. — Floating  bodies  in  their  dis- 
placement form  a  principal  wave,  to  which  J.  Scott  Rus- 
sell (who  has  given  much  time  to  these  researches,  and  to 
whom  we  owe  some  important  improvements  in  the  con- 
struction of  fast  sailing  boats,)  has  given  the  name  of  the 
great  wave,  or  the  solitary  wave.  This  wave  spreads 
more  or  less  upon  the  sides,  according  as  its  highest  point 
is  more  or  less  near  the  middle  of  the  length  of  the  body, 
25 


386  EESISTANCE   OF  FLUIDS. 

and  according  as  the  ratio  of  the  width  of  the  body  to  that 
of  the  canal  is  greater  or  less.  Upon  ordinary  canals  it 
forms  a  swell  whose  highest  point,  when  it  is  near  the 
middle  of  the  length  of  the  boat,  rises  from  0.65ft.  to 
0.98  ft.  above  the  general  level  of  the  canal ;  but  as  it  is 
found  farther  forward,  the  wave  shortens  and  rises  some- 
times 2.95  ft.  above  the  level  of  the  canal,  forming  there  a 
true  prow  wave,  in  which  the  bow  of  the  boat  seems 
to  be. 

We  may  readily  conceive,  that  the  form,  development, 
and  position  of  this  wave  must  exert  a  great  influence 
upon  the  intensity  and  upon  the  laws  of  the  resistance, 
and  thus  become  important  subjects  for  investigation. 

J.  Scott  Russell  inferred  from  observations  that  the 
velocity  of  propagation  of  the  solitary  wave  was  always 
equal  to  that  corresponding  with  half  the  depth  of  the 
water  in  the  canal,  increased  by  the  height  of  the  wave 
itself.  Now,  in  order  that  the  wave  shall  maintain  the 
same  position  relative  to  the  length  of  the  floating  body, 
and  that  the  resistance  shall  follow  a  regular  and  normal 
law,  the  boat  must  navigate  with  a  velocity  equal  to  that 
of  the  propagation  of  the  wave;  in  other  words,  the 
velocity  of  the  boats  must  be  regulated  by  the  depths  of 
water  in  the  canal  which  virtually  amounts  to  forbidding 
navigation  in  very  deep  water.  In  fact,  the  greatest 
speed  to  be  obtained  with  horses  exerting  an  inconsidera- 
ble effort  hardly  reaches  from  14.76  to  16.40  ft.  per  second, 
which  answers  to  heights  of  3.51  ft.  and  4.2  ft,  and  con- 
sequently, according  to  the  law  of  J.  S.  Russell,  to  depths 
of  7.02  ft.  and  8.49  ft.,  from  whence  it  follows  that  beyond 
these  depths,  the  navigation  of  these  boats  would  not  be 
possible. 

Inversely,  upon  canals  with  small  depths,  the  velocity 
of  the  boat  must  be  limited  so  as  to  lose  for  this  kind  of 
carriage  the  advantage  of  speed. 

It  appeared  necessary  that  I  should  make  various  ex- 


RESISTANCE   OF   FLUIDS.  387 

periments  upon  this  preliminary  question,  which  I  did 
first  at  Metz,  taking  advantage  of  the  favorable  disposi- 
tion afforded  by  the  long  trench  of  Fort  Saint  Vincent, 
and  afterwards  upon  the  Canal  de  1'Ourcq. 

When  the  boat  had  acquired  a  uniform  velocity,  the 
draught  suddenly  stopped,  the  motion  of  the  boat  slack- 
ened, the  wave  spread,  and  passed  on  in  virtue  of  its  own 
velocity  of  propagation,  which  was  observed  from  the 
bank,  by  means  of  marks  and  a  time-piece  indicating 
tenths  of  seconds.  We  may  conceive  that  these  observa- 
tions, in  making  which  it  was  difficult  to  seize  the  true 
time  of  the  passing  of  the  culminating  point  of  the  wave, 
are  not  very  precise. 

We  shall  see,  by  the  results  given  in  the  following 
table,  notwithstanding  the  difficulty  of  precise  observa- 
tions, that  there  existed  between  the  velocities  of  the  boat 
and  of  the  wave,  at  different  depths  and  drafts  of  water,  a 
nicety  of  agreement  sufficient  to  admit  that  the  velocity 
of  the  propagation  of  the  solitary  wave,  is  sensibly  the 
same  as  that  of  the  translation  of  the  boat  which  pro- 
duces it. 


388 


BESISTANCE   OF  FLUIDS. 

dd 


III 

&* 


It  4 


it 


II? 


«S  od  »-J  IH 


O  00  N  »O  Oi  O 
?O  T»<  r-t  t^  t-  •* 

05  rH  CO  <M'  -^  «0 
i—  1  rH  iH  rH  TH 


JH 

I 

5 


OCOrH 


I*' 


.  OS  t^  <N  T^ 

«S  od  o  co  w  co  co 


.         -      r  r-          " 

«S  ^'  00  0>  O  i-!  U5  »0 


(M  N  -*  W5  t>  CO  (N 

.  00  »O  O  rH  b-  OS  OS 


RESISTANCE   OF   FLUIDS. 

313.  Results  of  experiments  upon  the  resistance  of 
boats  to  towing. — After  this  preliminary  examination  of 
the  circumstances  of  the  phenomena,  we  give  the  results 
of  direct  experiments  upon  the  intensity  of  the  resistance 
in  its  ratio  to  the  velocity  of  motion. 

All  the  experiments  made  with  flat-bottomed  boats,  of 
five  different  forms  or  proportions,  have  shown  that,  on 
account  of  the  gradual  increase  of  the  longitudinal  incli- 
nations of  the  boat,  the  resistance  increases  much  faster 
than  the  square  of  the  velocity.  Furthermore,  if,  keeping 
account  of  the  observed  inclination,  we  determine  for 
eacli  case,  the  projection  of  that  part  of  the  boat  which 
lies  under  the  line  of  floatation,  upon  a  plane  perpendicu- 
lar to  the  direction  of  motion,  or  the  area  of  the  immerged 
section,  and  introduce  it  into  the  formula,  we  shall  still" 
find  that  the  ratio  of  the  resistance  to  the  square  of  the 
velocity  does  not  remain  constant,  so  that  it  does  not  seem 
possible,  in  this  case,  to  assign  any  simple  law  of  resist- 
ance. 

314.  fast  boats. — But  when  the  boat  presents  a  sharp 
prow,  nearly  vertical,  and  forms  that  cleave   the   water 
easily,  in  proportion  to  the  advance  of  the  boat,  the  resist- 
ance follows  these  laws  with  the  greater  closeness.    "When- 
ever the  speed  is  well  regulated,  and  the  principal  wave 
is  spread  upon  the  sides  of  the  boat,  so  that  the  latter  re- 
mains nearly  horizontal,  the  numerous  experiments  which 
I  have  made  with  many  boats,  upon  the  Canal  de  1'Ourcq — 
the  boats  being  constructed  after  the  model  of  those  of  the 
Paisley  canal,  in  Scotland — prove  that,  from  velocities  of 
towing  by  men  at  a  walk,  up  to  those  of  a  gallop  of  14.76  ft. 
and  upward,  the  resistances  follow  the  law  of  the  square 
of  the  velocities. 

A  graphic  representation  of  the  results  of  experiments, 
made  by  taking  the  squares  of  the  velocities  as  abscisses, 
and  the  efforts  exerted  for  ordinates,  gives  all  the  points 


390 


RESISTANCE  OF  FLUIDS. 


thus  determined  upon  a  straight  line,  which  cuts  the  line 
of  ordinates  above  its  origin.  This  circumstance  shows 
that,  in  this  case,  as  in  that  of  wheels  with  plane  paddles, 
the  resistance  is  composed  of  two  terms,  the  one  constant 
and  independent  of  the  velocity,  and  simply  proportional 
to  the  area  of  the  wetted  surface,  and  the  other  propor- 
tional to  the  square  of  the  velocity  and  the  area  of  the 
irnmerged  section.  But  the  first  term  is  always  so  small 
that  it  may  be  neglected  in  practice,  especially  in  all  cases 
of  high  velocity. 

Fig.  115,  which  represents  the  general  results  of  the 


FIG.  115. 


9         12         15        18         21 
Squares  of  Velocities  in  sq.  yds. 


24 


27       30 


1st,  2d,  5th,  6th,  9th,  10th,  13th,  14th,  15th,  and  16th 
experiments  made  in  1838,  upon  the  Canal  de  1'Ourcq,  in 
the  district  of  Meaux,  affords  an  example  of  this  law.  We 


RESISTANCE   OF  FLUIDS.  391 

see  from  a  general  inspection  of  all  the  resistances  meas- 
ured by  the  dynamometer,  when  the  boat  remained  hori- 
zontal, that  they  are  represented  by  a  straight  line  cutting 
the  axis  of  ordinates  or  of  resistances  in  a  point  which 
indicates  that  the  constant  resistance  was  about  16.75  Ibs. 
There  are  seen  upon  this  figure  a  certain  number  of  points 
marked  O  widely  separated  from  the  straight  line,  show- 
ing that  they  correspond  to  anomalous  cases.  In  fact, 
all  these  points,  which  answer  to  velocities  of  from  7.34  ft. 
to  10.8  ft.,  or  to  a  moderate  trot,  express  the  observed 
resistances  at  a  time  when,  by  the  displacement  of  the 
wave,  the  latter  was  found  in  front  of  the  boat,  which  was 
inclined  and  deeply  submerged  towards  that  part. 

The  following  table  contains  the  results  of  all  the  ex- 
periments which  I  made  upon  the  Canals  de  1'Ourcq,  and 
of  Saint  Denis,  with  three  models  of  boats.  The  most 
numerous  refer  to  the  model  of  fast  boats,  which  have 
been  a  long  while  in  service  between  Paris  and  Meaux. 
The  draft  of  water  of  these  boats  varied  from  0.9ft.  to 
1.4ft.,  and  their  displacement  from  202  c.  ft.  to  344.6  c.  ft. 
The  experiments  were  made  upon  the  ascent  and  descent, 
and  the  observed  resistances  were  compared  with  the 
results  of  the  formula 


392 


RESISTANCE   OF   FLUIDS. 


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RESISTANCE   OF   FLUIDS.  393 

315.  Consequences  of  the  experiments. — From  a  gen- 
eral view  of  these  results  we  may  infer  that  the  resistance 
to  towing  of  the  boat  upon  trial,  is  represented  with  all 
the  exactness  requisite  for  practice  by  the  formula 

K=0.38A'+16.04A(V±'y)2,  units  of  yds. ; 
or 

K=0.04301  +  0.1985A(V±0)',  units  of  feet; 

and  if  we  neglect  the  term  0.04301A',  independent  of  the 
velocity,  and  which  seldom  exceeds  15.4  or  17.6  Ibs.,  the 
formula  will  be 

R=0.19S5(V±?;)3  units  of  feet. 

This  value  of  the  resistance  to  towing,  in  the  most 
favorably  constructed  boat,  in  narrow  canals  with  small 
depths,  is  much  greater  than  that  experienced  by  sea 
vessels  in  deep  water,  which,  according  to  the  usual  esti- 
mate of  naval  engineers,  seems  to  correspond  with  a  value 
of  the  coefficient  K',  equal  to  4.6  Ibs.  or  6.2  Ibs.  per  sq.  yd.  • 
of  area  of  midships  for  a  velocity  of  1  yard  per  second. 

316.  Accidental  variations  of  the  resistance. — The  pre- 
ceding results  refer  to  cases  where  the  navigation  was  in 
a  normal  condition,  without  any  marked  disturbance  in 
the  position  of  the  wave  along  the  sides  of  the  boat,  which 
thus  preserved  nearly  a  horizontal  position.     But  when,  by 
accident,  the  horses  slackened  their  efforts  and  speed,  the 
motion  of  the  boat  was  momentarily  reduced ;  the  wave, 
which  had  a  velocity  of  propagation  equal  to  the  previous 
velocity  of  the  boat,  advanced  towards  the  prow  with  a  dif- 
ferential motion,  at  the  same  time  shortening  more  and 
more,  and  raised  the  bow, — which  was  found  to  be  deeply 
submerged, — inclined  the  boat,  and  arrived  at  the  prow, 
forming  a  kind  of  watery  hill  about  6.56  ft.  at  the  base  by 
2.62ft.  and  2.95ft.  in  height. 

We  may  conceive  that  in  circumstances  so  anomalous, 


394: 


KESISTANCE  OF  FLUIDS. 


the  resistance  must  increase,  though  the  velocity  might 
be  lessened,  and  then  we  might  truly  say,  that  the  resist- 
ance at  a  slow  trot  was  greater  than  at  a  smart  trot  or 
gallop.  The  following  results  show  how  effectually  the 
position  of  the  wave  exerts  an  influence  upon  the  in- 
tensity of  the  resistance  at  an  equal  velocity  : 

Comparison  of  the  resistance  to  towing  of  mail-boats,  when 
the  wave  is  spread  along  the  sides  and  when  it  is  towards 
the  bow. 


Kesistance  when  the 

Displacement 

wave  is 

of  the 
boat. 

Portion  of  the  canal  passed  over. 

Velocity. 

A 

oward  the 

toward 

middle  of 

the 

the  boat. 

bow. 

*tons. 

ft. 

Ibs. 

Ibs. 

5.716     ) 
7.147     f 

8.20 
14.10 

150 
211.7 

220  to  264 
397  to  441 

^rom   the  circular  basin  to  the 

6.23 

110.2 

265 

basin  of  Bondy 

6.56 

132.3 

253 

(ascending.) 

13.44 

99.2 

390 

9.758 

do 

6.23 

176.4 

297 

87KQ        j 

From  the  basin  of  Bondy  to  the 

14.43 

251.3 

421 

.  4  OO 

circular  basin  (descending)  

13.89 

264.6 

406 

r 

13.79 

240.3 

615 

8.758     J 

do 

14.43 
14.79 

266.8 
280. 

609 
600 

I 

15.14 

295.5 

565 

*  The  ton  here  =1000^.= 2205.48  Ibs. 


From  these  results  we  see  how  important  it  is,  in  this 
kind  of  navigation,  to  maintain  a  regular  speed,  and  it  is 
because  the  slow  trot  is  less  stable  and  more  liable  to  va- 
riations than  a  very  brisk  gait,  and  from  the  perturbations 
in  the  position  of  the  wave  being  more  frequently  pro- 
duced in  a  slow  pace,  that  we  find  for  it  a  greater  resist- 
ance than  for  a  gallop.  But  this  is  not  true  where  there 
is  no  disturbance  of  the  wave. 

"We  have  asserted  that  in  the  exceptionable  cases, 


RESISTANCE   OF   FLUIDS.  395 

where  the  wave  is  wholly  in  front  of  the  boat,  that  the 
resistance  goes  on  increasing  so  that  the  wave  cannot  be 
so  traversed  as  to  replace  the  boat  in  its  normal  position ; 
this  is  not  exactly  so.  It  has  frequently  happened  that  a 
boat  loaded  with  9.67  tons,  including  its  own  weight,  has 
raised  a  wave  in  front  2.95ft.  in  height,  and,  after  having, 
in  this  extraordinary  situation,  run  a  distance  of  656  ft.  to 
984ft.,  has  surmounted  the  wave  and  brought  itself  back 
to  a  horizontal  position,  and  the  wave  to  the  middle  of  its 
length.  Then  it  was  true  to  say,  that  the  resistance  was 
less  for  a  velocity  of  16.4ft.  than  for  one  of  13.84ft. ;  for 
in  the  first  case,  the  boat  moving  horizontally,  experi- 
enced, in  descending  from  Bondy  to  the  circular  basin,  a 
resistance  of  331  Ibs.  only,  while  in  the  second,  when  the 
prow  was  plunged  in  the  wave,  it  met  with  a  resistance 
of  617  Ibs.  But  this  difference  is  due  entirely  to  that  of 
the  circumstances  of  the  phenomena. 

317.  Recapitulation. — We  see  by  these  experiments, 
for  which  it  seemed  to  me  proper  to  enter  somewhat  into 
the  details,  that  when  the  form  of  boats  is  suitably  de- 
termined, so  that  their  position  in  relation  to  the  surface 
shall  experience  no  sensible  change,  the  resistance  follows 
the  law  of  the  square  of  the  velocity,  and  that  conse- 
quently the  fatigue  of  horses  employed  in  rapid  towing 
must   be  very  considerable.     We   are   thus   obliged   to 
shorten  greatly  the  length  of  the  relays,  and,  notwith- 
standing this  precaution,  we  still  lose  a  great  number  of 
horses. 

318.  Work  developed  ly  horses  in  hauling  fast  boats. — 
It  follows  from  the  experiments,  or  from  the  formula  ex- 
pressing their  results,  that  supposing  the  boat  has  only  60 
passengers,  and   goes,  for    example,  in    the    district  of 
Meaux,  where  the  velocity  of  the  water  is  v— 0.984ft.  at 
a  velocity  of  13.779  ft.  in  1",  or  9.32  miles  the  hour  on 


396  RESISTANCE   OF   FLUIDS. 

the  ascent,  and  at  that  of  14.107ft.  per  second,  or  9.62 
miles  per  hour  on  the  descent,  the  total  resistance  sur- 
mounted by  the  3  horses  would  be,  since  A— 6. 5124  sq.  ft. 

On  the  ascent, 

K=.1985  x  6.5124  (13.779ft-  +  0.984ft-)2=281.74lb3- 
On  the  descent, 

K=.1985  x  6.5124  (14.107-0.984)2=222.62lb3- 

or,  per  horse, 

On  the  ascent,  93.91lbs- 

On  the  descent,  74.2lbs- 

Consequently,  the  work  developed  by  each  horse  in  V  is 
as  a  mean,  in  this  case, 

On  the  ascent,  93.91lbs- xl3.779ft-=1293.98lbs-ft- 
On  the  descent,  74.2lbs-  x  14.107ft— 1046.74lbs- ft- 

Now,  from  the  results  of  direct  experiments  upon  the 
work  developed  by  horses  employed  in  other  modes  of 
transportation,  some  of  which  are  inserted  in  the  follow- 
ing table,  we  see  that  horses  employed  in  hauling  fast 
boats  develop  per  second,  during  their  service,  a  quantity 
of  work  more  than  triple  as  a  mean  of  that  of  the  carriage 
horse,  and  equal  to  one  and  a  half  times  that  of  the  dili- 
gence horse,  which  occasions  excessive  fatigue,  producing 
diseases  of  the  lungs,  of  which  they  nearly  all  die.  In 
exceptional  cases,  where  the  wave  is  in  front,  we  have 
said  that  at  a  velocity  of  13.84  ft.  the  resistance  has  some- 
times equalled  617  pounds,  which  exacts  from  each  horse 
an  effort  of  205.66  pounds,  and  the  excessive  work  of 
205.66lbs-  x  13.84ft-=2846lbs-ft-  in  V,  during  a  time  of  more 
than  one  or  two  minutes,  whence  results  straining  of  the 
hams  and  other  accidents. 


RESISTANCE   OF   FLUIDS. 


397 


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398  BESISTANCE   OF   FLUIDS. 

319.  Observation  upon  the  daily  work  of  horses. — 
These  examples  show  how  the  work  developed  by  animal 
motors  may  vary,  but  at  the  same  time  they  enable  us  to 
see,  that  when  we  exact  but  for  a  short  period  an  unusual 
work,  it  is  at  the  sacrifice  of  the  daily  work  which  may 
be  obtained  from  animals,  without  fatiguing  them  beyond 
measure,  or  speedily  ruining  them.    Thus  in  the  service 
of  the  mail-boats  from  Paris  to  Meaux,  the  distance  run 
for  each  relay  was  at  a  mean  12375.5  ft.,  which  was  accom- 
plished twice  a  day  by  horses,  on  the  ascent,  and  twice  a 
day  at  the  descent,  and  consequently,  according  to  the 
values  previously  found  for  the  resistance,  the  day's  work 
of  a  horse,  in  the  district  of  Meaux,  was  : 

At  the  ascent,  93.91  x  2  x  12375.5=2324366lb3- ft 
At  the  descent,  74.2  x  2  x  12375.5=1836524lb9- ft- 

Total    .     .     .    4160890lbs-ft- 

But  as  each  relay  was  accomplished  by  four  horses,  one 
of  which  rested  for  four  days,  the  mean  daily  work  was 
but  0.75  of  the  preceding  result,  or  equal  to 

3120667lbs-ft-, 

while  the  table  of  E~o.  318  shows  us  that  by  the  other 
modes  of  transportation,  and  without  an  excessive  fatigue, 
which  quickly  ruins  the  horses,  we  may  obtain  as  a  mean 
for  a  day's  work  of  a  draught  horse  12758606  pounds-feet, 
that  is  to  say,  a  work  four  times  as  great  as  that  obtained, 
with  considerable  loss,  from  horses  used  for  hauling  the 
mail-boats  of  the  Canal  de  1'Ourcq. 

320.  The  resistance  of  water  to  the  ^notion  of  wheels 
with  plane  paddles. — We  use,  to  transmit  motion  to  steam- 
boats, wheels  with  plane  paddles,  which,  impinging  upon 
and  pressing  the  water,  experience  a  resistance  which  is 
precisely  the  motive  power  by  means  of  which  the  boat 


RESISTANCE   OF  FLUIDS.  399 

is  impelled.  Direct  experiments  for  ascertaining  the  laws 
and  determining  the  intensity  of  the  resistance,  seemed 
to  me  necessary,  and  I  made,  in  1837,  many  experiments, 
of  which  I  give  a  succinct  analysis. 

For  these  experiments  we  employed  two  models  of 
wheels,  the  one  3.31  ft.  diameter  at  the  crown,  received 
at  will  the  paddles,  variable  in  number,  up  to  twenty  at 
most.  The  paddles  used  upon  this  wheel  had  successively 
for  their  dimensions : 

In  width  parallel  to  the  axis, 

0.32Sft-,     0.656"-,     0.984:ft-,     1.968ft- 

In  the  direction  of  the  radius, 

0.328f%     0.659ft-,     1.148ft-,     0.659ft- 

The  shaft  of  the  wheel  formed  a  windlass  around  which 
rolled  a  cord,  which  passed  to  the  summit  of  a  crane, 
55.77ft.  in  height,  which  supported  a  box  in  which  was 
placed  the  motive  weight.  The  wheel  was  established 
upon  a  fixed  frame,  and  the  depths  of  immersion  were 
varied  at  will,  in  raising  or  lowering  the  level  of  the 
reservoir  in  which  we  operated,  and  which  had  dimen- 
sions indefinite  in  relation  to  those  of  the  wheel. 

The  velocities  of  rotation  of  the  wheel  varied  from  the 
smallest  in  which  it  was  possible  to  observe  a  regular  mo- 
tion up  to  19.68ft.  per  second.  They  were  observed, 
when  the  motion  had  become  uniform,  by  means  of  a 
Breguet  timepiece,  indicating  tenths  of  seconds. 

The  whole  apparatus  was  so  arranged  as  to  reduce,  as 
much  as  possible,  the  passive  resistances  arising  from  the 
friction  of  the  axles,  the  rigidity  of  the  cord  and  the 
displacement  of  the  air,  and  a  reckoning  made  in  the  cal- 
culation of  the  results,  by  simple  formulae,  whose  details  it 
would  be  superfluous  to  publish. 

The  second  wheel  employed  was  8.567ft.  in  exterior 
diameter,  with  paddks  2.29  ft.  wide  in  the  direction  of  the 


400 


RESISTANCE   OF   FLUIDS. 


axis,  by  1.659  ft.  in  the  direction  of  the  radius  in  which 
they  were  placed.  The  depth  of  immersion  of  these  pad- 
dles was  successively  1.659  ft.,  1.325  ft,  and  0.937ft. 

For  each  number  of  paddles,  and  each  depth  of  im- 
mersion, the  motive  weights,  and  consequently  the  veloci- 
ties, were  gradually  changed,  so  as  to  have  a  series  of 
experiments  in  which  one  element  only  was  variable. 

Having  thus,  for  each  case,  the  values  of  the  resist- 
ance corresponding  with  the  different  velocities,  we  made 
a  graphic  representation  of  all  the  results  in  taking  for' 
abscissae  the  motive  weights,  and  for  ordinates  the  squares 
of  the  velocities  of  the  middle  point  of  the  immersed  sec- 
tion. 

In  all  the  series  so  represented  we  observed  that  up  to 
a  certain  velocity,  which  we  shall  indicate  hereafter,  all 

the  points  were  always 
(Fig.  116)  upon  a  straight 
line,  which  cut  the  line 
of  abscissae  in  front  of  the 
origin  at  a  point  O,  vari- 
able for  each  curve,  which 
shows  that  the  abscissas  or 
the  resistance  was  in  each 
case  represented,  as  for 
boats,  by  an  expression 
of  the  form 


O 


FIG.  116. 


calling  always 

A  the  immersed  surface  of  the  paddles ; 

Y  the  velocity  of  the  middle  submerged  section  of  the 
paddle ; 

K,  and  K/  constant  coefficients. 

The  immersed  surface  A  of  the  paddles  was  determined 
from  the  number  of  paddles  simultaneously  submerged, 
in  whole  or  in  part,  by  calculating  the  sum  of  the  im- 
mersed portions  of  the  floats  for  many  successive  positions 


RESISTANCE   OF   FLUIDS 


401 


of  the  wheel,  and  taking  the  mean  of  the  sums  of  surfaces 
thus  obtained.  It  thus  really  represents  the  mean  value 
of  the  total  surface  of  the  paddles  acting  upon  the  water. 
The  mode  of  representation  of  iigure  116  has  given  us  the 
value  of  the  constant  coefficient  K/ ;  since  the  abscissae 
AO  of  the  point  O  of  the  straight  line  expressing  the  law 
of  resistance  was  that  of  the  term  K/A.  It  is  thus  we 
have  obtained  the  following  values  : 


Dimensions  of  paddles 

Total  surface 

Constant 

resistance 

20  in  number. 

submerged. 

derived  from  the  trace 

KVA. 

per  square  feet 
K!*. 

ft. 
0.65  by  0.65 
0.98  by  1.15 
0.97  by  0.65 

sq.  ft. 
1.4693 
4.677 
4.4241 

Ibs. 
0.2867 
0.88219 
0.86013 

Ibs. 
0.19513 
0.18862 
0.19442 

Menu... 

0  19272 

The  trace  enabled  us  to  supply  the  value  of  the  coeffi- 
cient Kj  of  the  term  proportional  to  the  square  of  the 
velocity,  since  the  inclination  of  the  straight  line  express- 
ing the  law  of  resistance  is  given  by  the  expression 

E-K/A 


R— K/A  being  the  value  of  the  abscissae  of  this  straight 
line  diminished  by  AO,  and  Y2  being  that  of  the  ordi- 
nates.  Dividing,  in  each  case,  the  values  of  KXA,  given 
by  the  experiment  by  the  known  surface  A,  we  obtain 
the  values  of  the  coefficient  Kr 

321.  Causes  which  alter  the  law  of  resistance. — But, 
before  giving  the  values  of  the  coefficient  K^  of  the  resist- 
ance, furnished  by  a  summary  of  the  experiments,  we 
should  point  out  a  circumstance,  which,  in  altering  the 
26 


4:02  RESISTANCE   OF   FLUIDS. 

conditions  of  the  phenomena,  exerts  a  considerable  influ- 
ence upon  the  results.  In  order  that  with  different  veloci- 
ties and  depths  of  immersion,  the  wheel  and  its  floats  may 
be  in  comparable  conditions,  it  is  necessary,  as  we  have 
hitherto  implicitly  admitted,  that  the  void  formed  by  the 
paddles,  which  have  driven  before  them  the  water  upon 
which  they  have  acted,  should  be  constantly  replaced,  so 
that  the  next  paddle  submerged  may  meet  the  same 
resistance.  Now,  in  observing  the  motion  of  the  return 
of  the  water  into  the  void,  we  readily  understand  that  the 
refilling  must  be  accomplished  by  the  flowing  of  the  sur- 
face, as  it  were  over  a  dam  at  the  sides,  and  that  a  certain 
time  is  required  for  its  operation.  If,  then,  the  wheel 
turns  so  rapidly  that  the  void  has  no  time  to  fill,  the  pad- 
dles no  longer  finding  the  same  quantity  of  water  to  drive 

Fio.  117. 


251bs. 


10    20    30     40     50    60     70     80    90  100    110  120 
Squares  of  Velocities. 


as  in  less  velocities,  the  circumstances  of  the  phenomena 
are  changed,  and  accordingly  the  law  of  resistance  must 


RESISTANCE   OF  FLUIDS.  403 

be  modified.  This  change  increasing  more  and  more 
with  the  velocity,  it  happens  that  the  paddles  meet  a  less 
amount  of  water,  which  may,  so  to  speak,  be  naught,  so 
that  finally  the  wheel  turns  in  air  instead  of  water.  All 
these  effects  are  perfectly  manifested  by  the  trace  repre- 
senting the  results  of  experiments,  as  we  may  see  by  an 
inspection  of  figure  117,  relating  to  a  series  of  experiments 
made  with  20  'paddles  submerged  0.344  ft. 

The  line  representing  the  law  of  resistance  is  at  first 
nearly  a  straight  line  prolonged  up  to  a  certain  velocity 
depending  upon  the  depth  of  the  immersion  and  the  dis- 
tance apart  of  the  paddles ;  but  beyond  this  velocity  it 
departs,  more  and  more,  showing  that  the  resistance  no 
longer  maintains  its  proportion  with  the  square  of  the 
velocity.  All  these  facts  are  highly  important  to  steam 
navigation,  for  they  show  that  it  is  necessary  to  establish 
between  the  depths  of  immersion  of  the  paddles,  their 
distance  apart,  and  the  velocity  with  which  they  are  im- 
pelled, such  relations  that  the  water  may  always  have 
time  to  fill  up  the  voids,  and  that  for  each  wheel  con- 
structed, there  is  a  limit  of  speed,  adapted  to  the  best 
effect. 

322.  Proper  distance,  of  the  paddles  apart. — Without 
going  into  further  details  upon  these  remarkable  effects, 
I  content  myself  with  saying  that  the  law  of  proportion- 
ality of  the  second  term  of  the  resistance  to  the  square  of 
the  velocity  has  been  verified  up  to  velocities  ot  5.444  ft. 
and  6.232  ft,  when  the  spaces  of  the  paddles  on  the  outer 
circumference  of  the  wheel,  were  comprised  within  from 
two  to  three  times  their  depth  of  immersion.     This  pro- 
portion, moreover,  is  conformable  to  the  ordinary  practice. 

323.  Value  of  the  coefficient  Kt  of  the  second  term  of 
the  resistance. — Having  regard  to  the  circumstances  which 
we  have  pointed  out,  and  consequently  restricting  the  law 
of.  the  resistance  within  the  limits  of  our  ability  to  verify 


404: 


RESISTANCE    OF   FLUIDS. 


them,  we  will  make  known  the  results  relating  to  experi- 
ments in  which  the  velocity  of  the  wheel,  and  the  spaces 
of  the  paddles  allowed  a  complete  return  of  the  water. 

The  values  of  the  coefficient  E^  furnished  by  the  ex- 
periments were  as  follows  : 


Values  of  the  coefficient  K^  of  the  formula 
K=A(K1/+K)V!I). 


Number  and  dimensions  of  the  paddles. 

Values  of 
Kj. 

" 

10  paddles  0  33ft  by  033  ft              *  

1  990 

5       «           "                "       

2.0748 

Wheel  3  31ft 

10       "      0  66  ft         0  66  ft 

1  9319 

5       "           «               "       

2.0794 

5       "      098ft        1.15ft  

2.2365 

10       "       197ft        066ft  

1.9928 

5       «           «                "                     

2  2460 

Great  wheel 
8.567ft. 
in  diameter. 

fl.659 

8   paddles  of  2.29ft.  submerged...  -j  1.325 
(0.937 

General  mean... 

2.2269 
2.1698 
2.4078 

2.1355 

The  general  mean  does  not  differ  over  T\  from  the  partial 
results,  and  we  see  that  when  the  spaces  of  the  paddles 
were  within  the  indicated  limits,  that  the  effect  exerted 
by  wheels  with  plane  floats  upon  the  axle  may  be  repre- 
sented by  the  formula 

K=A  [.19272+2.13559V2], 

A  being  the  mean  of  the  surfaces  of  paddles  simulta- 
neously submerged  at  rest. 

V  the  absolute  velocity  of  the  wheel. 

324.  Case  where  the  wheel  turns  in  running  water. — 
The  wheel  which  had  served  for  the  above  experiments 
having  been  placed  in  a  small  wooden  canal,  3.7  ft.  wide 
by  2  ft.  deep,  the  same  course  was  taken  to  ascertain  the 


RESISTANCE   OF   FLUIDS.  405 

law  of  resistance.  Without  going  into  further  details,  I 
will  simply  state  that  the  results  of  these  new  experiments 
are  also  represented,  with  sufficient  exactness  for  practice 
by  the  same  formula,  by  adding  or  subtracting  the 
velocity  v  of  the  current  to  or  from  that  of  Y,  so  that  the 
general  expression  of  effort  exerted  upon  still  or  running 
water,  by  the  paddles  of  wheels  with  plane  floats  will  be 

R=A  [0.1927+2.0785  (Y±^)9], 

in  taking  for  the  coefficient  of  the  second  term  a  number 
which  conforms  best  to  all  the  experiments. 

325.  Influence  of  the  presence  of  a  boat  near  the 
wheels. — The  experiments  in  question  were  made  upon 
isolated  wheels,  and  it  was  proper  to  ascertain  whether 
the  presence  of  a  boat  near  the  wheel  would  exert  any 
influence  upon  the  intensity  and  law  of  the  resistance. 

For  this  purpose,  we  placed  near  the  wheel,  at  a  dis- 
tance of  0.13ft..  parallel  to  the  exterior  vertical  plane  of 
the  floats  of  the  wheel,  8.567  ft.  in  diameter,  a  boat  sub- 
merged an  equal  depth  with  the  floats,  and  made  two  sets 
of  experiments,  with  depths  of  immersion  of  1.325  ft.  and 
0.84  ft.,  to  compare  the  results  with  those  of  the  series 
made  in  the  case  when  the  wheel  was  isolated,  and  its 
paddles  immersed  1.325  ft.  and  0.937  ft. 

The  results  of  these  experiments  seem  to  show  that  by 
reason  of  the  obstacle  which  the  presence  of  the  boat  op- 
posed to  the  return  of  water  into  the  void  formed  by  the 
float,  the  resistance  diminished  somewhat,  but  so  small  a 
quantity  that  it  may  come  within  the  limits  of  the  errors 
of  observation.  In  fact,  we  found 

At  the  depth  of  immersion  of  1.325  ft.  without  boat  Ki=2.1698 

"  "  "  "  "  "  '«  with  boat  K,=2.1413 

At  the  depth  of  immersion  of  0.937ft.  without  boat  KI  =2.4078 

"  "  "  "  «  "0.84ft.  with  boat  K!  =-2.1580 


406  RESISTANCE    OF   FLUIDS. 

"We  see,  then,  that  the  preceding  formula  derived  from 
a  summary  of  the  experiments  may  be  still  applied  to  the 
case  where  the  wheel  is  placed  at  the  side  of  a  steamboat. 

326.  Application  to  the  wheels  of  steamboats.  —  The 
formula  of  the  resistance  experienced,  and  the  effect  trans- 
mitted by  the  paddles  of  a  wheel  turning  in  water  being 


when  the  axis  of  this  wheel  has  no  motion  of  translation, 
it  is  clear  that  if  this  axis  is  borne  upon  a  steamboat  going 
with  a  velocity  Y,  the  paddles  will  only  impinge  upon  the 
water  with  a  velocity  U—  Y,  and  that  in  this  case  the 
formula  expressing  the  resistance  experienced  by  the 

floats  will  be 

—  Y)2 


in  still  water,  and  finally,  that  if  the  boat  bearing  the 
wheel  ascends  or  descends  a  stream  running  with  a  veloc- 
ity v,  the  expression  of  resistance  will  be 


-Y—  v?  on  the  ascent, 
K^K.A  (TJ—  Y+  v)*  on  the  descent, 

If  we  examine  particularly  the  case  of  navigation  in 
still  water,  the  work  of  this  resistance,  or  that  of  the  ma- 
chine moving  the  wheel  in  \"  will  be 


and  if  we  express  in  horse  powers  of  550  lbs-  ft-  the  effect- 
ive force  of  the  motor  will  be 


550  550 

An  observation  of  existing  constructions  will  allow  us 
to  judge  whether  the  value  of  the  coefficient  K,  derived 


RESISTANCE   OF   FLUIDS. 


407 


from  the  experiments  above  reported,  agrees  with  the 
observed  facts  of  navigation.  Indeed,  we  have  for  each 
boat  the  dimensions  of  the  wheels  and  floats,  and  the 
number  of  the  latter,  from  which  we  can  deduce  the  sub- 
merged surface  of  the  paddles. 

Observation  gives  us  the  velocity  U  of  the  paddles, 
which,  by  reason  of  their  small  height  compared  to  the 
radius  of  the  wheels,  may  be  regarded  as  the  point  of  ap- 
plication of  the  resistance,  as  well  as  the  velocity  Y  of  the 
boat,  and  if  we  introduce  the  value  of  1^=2.13559, 
derived  from  our  experiments,  the  above  formula  should 
give  the  effective  force  of  the  machine,  such  as  observa- 
tion has  furnished.  Direct  experiments,  made  by  hauling 
upon  a  fixed  point,  in  giving  the  effort  exerted  and  trans- 
mitted by  the  paddles  to  set  the  boat  in  motion,  enable 
us  to  verify  directly  the  formula 

R=2.13559AU2, 

by  introducing  the  particular  data  of  each  case. 

In  making  this  comparison  upon  the  steamers  the 
Sphinx,  the  Mentor  of  160  nominal  horse-power,  the 
Medee,  and  the  Veloce  of  220  horse-power,  for  which  the 
dimensions  and  different  velocities  are  given  by  M.  Cam- 
paignac,  in  his  work  upon  steam  navigation,  we  have  the 
following  data  and  results  :* 


1*4 

l| 

o 

2£ 

Names  of 
steamers. 

Horse  power  of  each  of 
the  two  engines. 

N. 

•s&g 

ci^ 
«11 

*?| 

tl1P 

S-o  §< 

w 
** 

11 

1! 

^ 

%** 

ga 

IS 

>B 

nominal,     effective. 

sq.  ft. 

ft. 

ft. 

80          80  851 

41  227 

19.993 

15.190 

2.3074 

80        80.854 

37.136 

20.862 

15.528 

1.9897 

Medee  

110       110.380 

56.371 

20.823 

16.201 

2.4318 

V61oce 

110      111.350 

42.928 

20.948 

15.948 

2.6879 

2.3542 

*  This  table  was  calculated  for  the  French  H.  P.  of  75  kilometres,  or  543  Ibs.  ft. 


408  RESISTANCE   OF   FLUIDS. 

AVe  would  observe  that  the  value  of  the  whole  simul- 
taneously submerged  area  of  the  paddles  was  determined 
by  tracings,  and  on  the  supposition  of  the  vertical  floats 
being  entirely  submerged  a  little  below  the  surface,  but 
probably  less  than  it  was  in  reality,  so  that  the  values  of 
Kx  are  undoubtedly  greater  than  they  should  be.  It  is 
not,  then,  surprising  that  the  mean  of  these  values  sur- 
passes those  derived  from  our  direct  experiments. 

327.  The  resistance  of  the  air. — The  phenomena  pro- 
duced by  bodies  moving  in  air,  are  similar  to  those  pre- 
sented by  liquids,  and  the  resistance  which  it  opposes  to 
the  motion  of  these  bodies  is  of  the  same  kind.     Still,  it 
is  proper  to  distinguish  between  what  occurs  in  uniform 
motion  from  that  which  takes  place  in  variable  motion. 

In  the  first  case,  the  velocity  remaining  the  same,  the 
fluid  molecules,  successively  driven  aside  by  the  body, 
experience  the  same  displacements,  receive  the  same 
velocities,  and  in  different  instants  of  its  motion,  the  body 
meets  the  same  resistance.  But  in  variable  motion,  accel- 
erated, for  example,  the  fluid  molecules  receive  greater 
and  greater  degrees  of  velocity,  and  as  they  belong  to  an 
elastic  fluid,  the  fluid  prow  formed  in  front  of  the  body 
acquires  a  density  and  mass  continually  increasing,  whence 
it  follows  that  the  mass  displaced  increases  in  the  same  time 
as  the  velocity  imparted  to  it.  We  conceive  then,  a  priori, 

that  the  greater  the  acceleration  of  motion    ,  so  will  be 

t 

the  resistance,  and  so  we  may  foresee  that,  in  accelerated 
motion,  the  expression  of  the  resistance  of  the  air  must 
comprise,  besides  other  terms,  one  peculiarly  due  to  the 
acceleration  of  motion  itself.  It  was  reserved,  however, 
for  the  experiments  at  Metz  for  the  first  proving  of  this 
matter,  as  we  shall  see  anon. 

328.  Results  of  experiments. — The  celebrated  Borda 
made,  in  1763,  experiments  upon  the  laws  of  the  resist- 


RESISTANCE   OF   FLUIDS. 


409 


ance  of  air,  by  means  of  a  kind  of  fan- wheel,  with  a  ver- 
tical axle  and  horizontal  arms,  a  little  over  7.15ft.  in 
length.  He  placed  at  the  end  of  this  arm  the  surfaces 
and  different  formed  bodies  on  which  he  wished  to  oper- 
ate, and  he  observed  the  uniform  velocities  of  the  fly- 
wheel under  the  action  of  different  weights.  lie  thought 
the  influence  of  the  friction  of  this  apparatus  might  be 
overlooked,  which  has  occasioned  some  uncertainty  in 
his  results ;  for  it  is  difficult  to  admit  that  in  dealing  with 
so  small  a  resistance,  the  portion  of  the  motive  weight 
engaged  in  overcoming  the  friction,  should  not  be  com- 
parable to  that  surmounting  the  resistance  of  the  air. 

Borda  placed  in  succession  at  the  ends  of  the  arms  of 
his  apparatus,  square  surfaces  of  *9.56,  6.38,  and  4.25 
inches  at  the  sides,  and  set  them  in  motion  with  weights  of 
f8.8,  4.4,  2.2,  1.1,  and  0.5  pounds,  and  consequently  with 
different  velocities.  From  the  dimensions  and  data  rela- 
tive to  this  apparatus,  the  author  has  calculated  the  resist- 
ances of  the  air  corresponding  with  the  different  veloci- 
ties, and  the  results  expressed  in  yards  are  given  in 
the  following  table : 


Results  of  JB  or  da's  experiments  upon  the  resistance  of  air. 


Surface  of  9.591  in.  each  side, 
or  of  .07099  sq.  yds. 

Surface  of  6.894  in.  each  side, 
or  of  .08155  sq.  yds. 

Surface  of  4.268  in.  each 
side,  or  of  .01402  sq.  yds. 

Eesistance 
of  air. 

i 
** 

"3 

<M     . 

Resistance 
of  air. 

Velocities. 

Squares  of 
velocities. 

1 

^o 

Squares  of 
velocities. 

Ibs. 
0.16695 
0.07895 
0.04168 

0.02084 

yds. 
3.787 
2.690 
1.891 
1.334 

14.34 
7.237 
3.579 
1.780 

Ibs. 
.16713 
.08356 
.0416 
.02083 
.01036 

yds. 
5.938 
4.199 
2.978 
2.091 
1.382 

35.26 
17.64 
8.86 
4.28 
1.91 

Ibs. 
.1592 
.0796 
.0399 
.01986 
.00944 

9.05*3 
6.397 
4.505 
3.84 
2.252 

81.94 
40.93 
20.30 
10.14 
5.07 

*  9,  6,  and  4  "  ponces.' 


f  8,  4,  and  2  "livres." 


410  RESISTANCE    OF   FLUIDS. 

If  we  represent  these  results  graphically,  in  taking 
the  resistances  for  abscissae,  and  the  squares  of  velocities 
for  ordinates,  we  find  all  the  points  relative  to  the  same 
surface  are  situated  in  a  straight  line,  thus  indicating  that 
the  resistance  increases  as  the  square  of  the  velocity. 
The  small  extent  of  surfaces  used  by  the  author  could  not 
manifest  with  certainty  the  existence  of  a  constant  term, 
in  the  expression  of  resistance. 

Comparing  these  results  with  the  formula  R^K^AV2, 
(expressed  in  yards  and  square  yards)  we  have  for  Kj  the 
following  values : 

Square  of  9.585ln-  or  0.26575^-  per  side,  £,=0.1618.* 
Square  of  6.390in-  or  0.17716yd3-  per  side,  K, =0.1472. 
Square  of  4.263in-  or  0.1181  yds-  per  side,  1^=0.1382. 

It  is  to  be  observed  that  Borda  having  neglected  the 
influence  of  friction,  which  increases  with  the  resistance 
and  the  motive  weights  employed,  the  apparent  diminu- 
tion of  the  resistance  for  the  smaller  surfaces  may  be 
attributed  to  this  cause. 

329.  Experiments  by  M.  Thibault  upon  bodies  in  mo- 
tion in  the  air. — We  are  indebted  to  this  officer,  whom 
the  naval  service  lost  too  early,  for  numerous  and  very 
well  executed  experiments,  published  at  Brest  in  1832. 
M.  Thibault  used  for  his  experiments  a  fly-wheel  with 
two  wings,  turning  on  a  horizontal  axle,  and  moved  by  a 
weight,  which  the  resistance  of  the  air  itself  soon  rendered 
uniform.  This  very  light  wheel  was  composed  of  a  steel 
axle  2.13ft.  in  length  by. 016 ft.  square,  terminated  by 
journals  with  a  diameter  of  .0082  ft.  The  arms  of  the 
fly  were  each  formed  of  an  iron  rod  8.97  ft.  long  by  .045  ft. 
wide  in  the  direction  of  movement  near  the  axle,  and 
0.016ft.  near  the  ends,  with  a  constant  thickness  of 
0.019  ft.  in  a  direction  parallel  to  the  axle.  The  side  of 
the  arm  striking  the  air  was  bevelled. 

*  The  coefficient  K!  is  for  A  in  sq.  yds.  and  V  in  yds. 


RESISTANCE    OF   FLUIDS. 


411 


g  =  l 

IIP 

I  ill 


1" 


0>^3 

iil 

all* 

g  p. 


II 


If 


11 

fS 


cooocT5-<*<»-ia5cowiO-*'-<-«*<'-ioo-*iait- 


000 


o  o*  o* 


^v— 

t-  t^  »>  »>•  00  Ci  O<  O_  i-H  CO  1C  CO  CO  i-J  r-J  SO  O5 

s^  <N  <?4  <M'  <M'  s<i  eo  co  eo  co  co  co  -*'  10  o  t-'  os 


od  06 


412  RESISTANCE   OF   FLUIDS. 

The  wings  were  mounted  upon  the  arms  of  the  flyer, 
and  at  first  directed  in  planes  passing  through  the  axis, 
then  by  means  of  suitable  arrangements  they  were 
inclined,  1st,  in  turning  them  around  the  radius  ;  2d,  in 
turning  them  round  parallel  to  the  axis,  so  that  their 
direction  left  the  axis  either  in  the  front  or  rear.  The 
inclinations  thus  obtained  were  varied  at  intervals  of  five 
degrees,  and  were  carefully  measured.  The  motion  of 
the  fan-  wheel  was  produced  in  all  cases  by  the  same  mo- 
tive weight  of  8.82  pounds,  and  the  duration  of  20  turns 
made  with  uniform  motion  was  observed. 

I  have  discussed  and  calculated  the  results  of  the  ex- 
periments of  M.  Thibault,  in  applying  the  formula 


which  represents,  as  we  shall  see  hereafter,  all  the  results 
of  the  experiments  made  at  Metz.  In  giving  to  the  coeffi- 
cient K/  relative  to  the  constant  resistance,  independent 
of  the  velocity,  the  value  K^.08002,  (units  of  yards,) 
derived  from  our  experiments  upon  a  fan-wheel,  we  were 
enabled  to  deduce  the  value  of  the  coefficient  Kx  depend- 
ent upon  the  velocity.  Account  was  also  taken  of  the 
inclination  of  the  surface  of  the  wings  towards  the  direc- 
tion of  the  motion,  by  introducing  in  the  second  term  of 
the  formula,  in  place  of  the  area  Ar=0.12323  sq.  yds.,  its 
projection  upon  a  plane  perpendicular  to  the  direction  of 
the  motion. 

The  table  of  the  preceding  page  contains  the  data  of 
the  experiments  of  M.  Thibault,  and  the  results  of  the 
calculations.  The  figures  entered  in  this  table,  show 
that  the  resistance  per  square  yard  of  surface  projected 
perpendicularly  to  the  direction  of  the  motion,  and  per 
yard  of  velocity,  where  the  value  of  the  coefficient  Kx  of 
the  formula 


RESISTANCE    OF   FLUIDS.  413 

does  not  decrease  so  long  as  the  angle  of  inclination  is 
not  below  from  50  to  60°. 

330.  Remarks  upon  wing  regulators  and  wind-mills. — 
It  follows  in  the  case  of  fan  fly-wheels,  used  as  regulators 
of  motion,  where  the  wings  are  inclined,  and  turn  around 
the  radius  of  the  fly-wheel,  that  when  the  motive  power 
is  too  feeble,  we  do  not  have  a  diminution  of  resistance 
until  the  wings  have  passed  the  inclination  of  from  50  to 
60°,  and  as  these  regulators  should  also  serve  to  prevent 
the  acceleration  of  motion  when  the  motive  power  in- 
creases, and  consequently  then  afford  the  greatest  resist- 
ance, it  would  be  well,  in  the  normal  state,  to  place  them 
at  an  angle  of  about  35°  with  the  plane  perpendicular  to 
the  direction  of  the  motion. 

It  seems  to  me  that  something  analogous  to  this  is 
produced  in  wind- mills,  whose  sails  are,  by  some  special 
mechanism,  made  to  incline  when  the  wind  has  acquired 
too  much  intensity. 

Experience  shows,  in  fact,  that  this  disposition,  whose 
aim  is  to  check  the  velocity  from  being  too  greatly  accel- 
erated by  the  effect  of  squalls,  does  not  fully  attain  its 
object,  and  that  the  mill,  whose  normal  velocity  is  from  5 
to  6  turns  in  1  minute,  by  a  good  breeze  from  16  to  19  ft.  of 
velocity  per  second,  reaches  that  of  from  29  to  30  turns, 
and  more,  with  greater  winds. 

331.  Experiments  upon  different  formed  surfaces. — 
M.  Thibault  has  successively  repeated  the  same  experi- 
ments with  concave  cylindrical  surfaces ;  he  arrived  at 
the  same  consequences,  and  has  established  the  fact  that, 
with  an  equal  projection  of  surface,  upon  a  plane  perpen- 
dicular to   the  direction  of  the  motion,  the  resistance  in- 
creases with  the  curvature,  but  quite  gently. 

As  for  hollow  surfaces,  with  double  curvature,  such  as 
those  formed  by  canvas  fixed  upon  the  four  sides  of  a 


414  RESISTANCE    OF   FLUIDS. 

frame,  the  resistance  increases  with  the  curvature,  and 
more  rapidly  than  in  the  preceding  case. 

A  comparison  was  made  of  the  resistance  offered  by 
bent  sails,  with  that  experienced  by  plane  sails  with  the 
same  surface  as  that  of  the  sails  developed ;  the  two  sur- 
faces of  folded  sails  were  each  0.1302  sq.  yds.,  and  the 
lower  side  was  brought  near  the  upper,  as  is  usual  with 
sails  under  the  action  of  wind,  and  the  author  found  that 
the  resistance  of  the  bent  surface  was  the  same  as  that  of 
the  plane  surface,  notwithstanding  the  diminution  of  the 
projection  of  the  first  surface  upon  the  direction  of  the 
motion.  A  comparison  is  thus  made  between  the  increase 
of  the  resistance  due  to  the  curvature,  and  the  diminution 
due  to  the  narrowing  of  the  projected  surface. 

This  consequence  is  important,  inasmuch  as  it  facili- 
tates the  applications  relative  to  the  action  of  wind  upon 
the  sails  of  vessels. 

332.  Influence  of  the  inclination  of  the  wings. — The 
author  has  found  that  when  the  vanes  are  inclined  so  that 
the  axis  of  rotation  is  found  in  front  of  their  plane,  in 
regard  to  the  direction  of  motion  (Fig. 
118)  the  resistance  diminishes  rapidly 
as  the  inclination  increases,  and  that  at 
FIG.  us.  the  inclination  of  55°  it  is  not  more 

than  0.5T15  of  the  perpendicular  resistance,  while  when 
the  axis  of  rotation  is  found  behind  the  plane  of  the  wings, 
/  the  resistance  goes  on  increasing  even 

\/- =©  up  to  the  angle  of  55°,  (Fig.  119,)  for 

which  it  is  equal  to  1.2293  times  the 
perpendicular  resistance. 

These  results  show  that  this  mode  of  inclining  the 
vanes  of  fly-regulators,  answers  readily 
the   proposed  purpose,  since   in   dis- 
posing them  so  that  the  vanes  may  be 
inclined  at  will  in    either  direction, 
(Fig.  120,)  the  resistance  experienced  may  be  rendered 
greater  or  less,  according  to  the  necessities  of  the  case. 


RESISTANCE   OF  FLUIDS.  415 

The  same  experiments,  repeated  upon  curved  surfaces, 
with  different  degrees  of  inclination,  have  led  to  similar 
consequences,  while  indicating  a  still  greater  intensity  of 
resistance  than  is  experienced  by  plane  surfaces.  This 
explains  the  advantage  which  navigation  derives  from 
the  movements  of  rotation  impressed  upon  sails  parallel 
to  the  axis  of  the  masts. 

333.  Influence  of  the  approximation  of  the  surfaces 
exposed  to  the  resistance  of  the  air.—M..  Thibault  has  also 
made  some  experiments  to  ascertain  whether  two  equal 
surfaces  (placed  one  behind  the  other,  a  very  small  dis- 
tance apart)  experience  a  less  total  resistance  than  when 
isolated.     For  this  purpose  he  mounted   upon   his   fly- 
wheel four  wings,  placed  in  pairs,  the  one  behind  the 
other,  at  a  distance  which  he  has  not  given,  and  he  found 
for  the  case  upon  which  he  operated,  that  the  resistance 
of  the  posterior  was  not  over  f  of  that  of  the  anterior  sur- 
face.    This  result,  which  can  be  applied  to  railroad  trains, 
is  important,  and  it  is  very  desirable  that  more  complete 
experiments  should  be  made  upon  this  subject. 

334.  Influence  of  the  form  of  surfaces. — The  same  ex- 
perimenter having  placed  at  the  extremities  of  his  fly- 
wheel various  surfaces  of  the  same  area,  but  of  which  two 
were  square,  two  circular,  and  two  in  the  form  of  a  right- 
angled  triangle,  so  that  the  centre  of  their  figure  was  in  all 
cases  at  the  same  distance  from  the  axis,  Jias  observed 
that  under  the  action  of  the  same  motive  weight  the  fly- 
wheel took,  in  all  cases,  the  same  velocity,  which  shows 
that  the  resistance  is  independent  of  the  form  of  the  plane 
surfaces  experimented  upon. 

335.  Resistance  of  air    to    the    motion  of  spherical 
bodies. — This  particular  case,  which  is  of  special  interest 
in  the  study  of  the  motion  of  projectiles  in  the  air,  has  for 


416  RESISTANCE   OF   FLUIDS. 

a  long  time  occupied  the  attention  of  philosophers  and 
geometricians.  ISTewton  was  the  first  to  experiment  upon 
this  subject,  in  observing  the  fall  of  spherical  bodies. 
Hutton  and  other  observers  have  studied  this  resistance 
in  the  case  of  small  velocities,  by  means  of  a  rotating  ap- 
paratus, and  more  lately  the  latter,  in  comparing  the 
velocities  of  projectiles,  at  different  distances  from  the 
piece  of  ordnance,  has  extended  his  researches  to  great 
velocities.  Finally,  within  a  few  years  numerous  ex- 
periments upon  this  last  part  of  the  question  have  been 
made  at  Metz. 

We  limit  ourselves  to  indicating  the  results  more 
especially  applicable  to  industrial  questions. 

From  a  summary  of  JSTewton's  experiments  upon  the 
fall  of  glass  globes  in  air,  with  velocities  comprised  be- 
tween zero  and  29.528ft.  per  second,  at  a  mean  tempera- 
ture of  53.6°,  and  at  a  pressure  of  2.46  ft.,  the  value  of  the 
coefficient  K^  was  about  .0007137,  so  that  the  resistance 
experienced  by  spheres  moved  in  the  air,  at  velocities 
comprised  within  these  limits,  would  be 

K=0.0007137  AV2=0.0007137-5j-  V2,  for  units  of  feet  ; 


orK=0.05781  AV2=0.05781-^-V2,  for  units  of  yards  ; 

\.2tlo 

but,  in  great  velocities,  the  coefficient  of  the  resistance 
increases  with  the  velocity,  and  after  a  discussion  of  Hut- 
ton's  experiments,  and  those  of  the  Commission  at  Metz, 
General  Piobert  has  proposed,  for  a  representation  of  the 
law  of  the  resistance  of  the  air  to  the  motion  of  projec- 
tiles, the  formula 

K=0.03546  AV2(1+.  002103V),  units  of  yards, 
B=0.00043778  AV2(l-f  0.0070102V),  units  of  feet; 

which   would  indicate   that,  with   these  velocities,  the 


RESISTANCE   OF   FLUIDS.  417 

expression  of  the  resistance  must  contain  a  term  propor- 
tional to  the  cube  of  the  velocity,  and  that  the  constant 
term  is  without  a  sensible  influence. 

336.  Experiments  at  Metz  upon  ~bodies  moving  in  air. — 
Numerous  experiments  with  the  joint  labor  of  MM. 
Piobert,  Didion,  and  myself,  were  made  at  Metz  in  1835 
and  1837,  which  were  more  particularly  made  by  M. 
Didion,  in  which  we  made  use  of  chronometric  apparatus, 
similar  to  those  described  in  K~os.  225  and  77,  to  observe 
the  law  of  the  descent  in  air  of  different  formed  bodies, 
and  of  different  dimensions.  These  experiments  were 
made  at  the  ancient  foundry  of  Metz,  where  we  could 
avail  ourselves  of  a  vertical  fall  of  46.916  ft. 

The  bodies  employed  were  suspended  upon  a  silk  cord, 
wound  round  a  pulley,  which,  in  its  motion,  bore  a  style, 
whose  trace  upon  the  plate  of  the  chronometric  appa- 
ratus, impressed  with  a  known  uniform  motion,  and 
observed  at  every  experiment,  furnished  the  law  of  mo- 
tion of  the  descent  of  the  body. 

Special  experiments  were  made  to  determine  the  pas- 
sive resistances  of  the  apparatus  to  keep  an  account  of 
them  in  the  calculations. 

Without  going  into  a  detailed  discussion  of  the  results, 
and  the  tests  applied  to  them,  we  simply  indicate  the 
method  adopted  for  the  calculations. 

We  have  previously  seen  that  the  experiments  upon 
the  resistance  of  water  have  compelled  us  to  admit,  in  the 
expression  of  the  resistance  of  fluids,  the  existence  of  a 
constant  term,  and  that  of  a  term  proportional  to  the 
square  of  the  velocity.  This  conclusion  has  been  con- 
firmed by  the  experiments  which  we  have  made  upon 
the  resistance  of  air,  obtained  from  uniform  motion. 

A  first  series  of  experiments,  made  upon  a  thin  plate 
1.267  sq.  yds.,  gave  us  for  the  expression  of  the  resistance 
of  the  air, 

K=0lb9.0663  A+0.1372  AV2  units  of  yards ; 

27 


418 


RESISTANCE   OF  FLUIDS. 


but  as  the  fall  of  46.9  ft.  was  not  sufficient  to  give  at  the 
end  of  it  a  strictly  uniform  motion,  and  as  we  shall  pres- 
ently see  that  the  resistance  in  variable  motion  must 
comprise  a  third  term  dependent  upon  the  acceleration 

-  of  motion,  it  follows  that  the  term  0.13T2  AY2,  which 
t> 

comprises  implicitly  this  third  term  is  a  little  too  great, 
and  should  be  diminished. 

The  existence  of  a  constant  term  in  the  expression  of 
the  resistance  was  manifested  in  the  experiments  made 
upon  a  wheel  with  wings  1.09  yd.  internal  diameter,  bear- 
ing square  wings  0.2187  yd.  by  0.2187yd.,  20  in  number, 
presenting  thus  a  total  surface  of  0.9568  sq.  yds.  The 
results  of  these  experiments  were  very  exactly  represented 
in  the  case  of  uniform  motion  by  the  formula 


and 


K=0lbs.008892  A+0.001907AY8,  in  units  of  feet, 
K=0lbs.08002  A+0.1548  AY2,  in  units  of  yards, 


as  may  be  seen  in  the  following  table,  in  which  the  values 
found,  at  different  uniform  velocities,  for  the  coefficient 
of  the  term  proportional  to  the  square  of  the  velocity,  are 
very  nearly  constant. 


Experiments  upon  the  resistance  of  air  to  the  motion  of  a 
wheel  with  plane  plates. 


Uniform  velocity  of  the  centre  of 
resistance  of  wings  in  yards  per 
second, 

yds. 
2.89 

yds. 
4.11 

yds. 
5.17 

yds. 
5.89 

yds. 
6.69 

yds. 
7.20 

yds. 
7.83 

yds. 
8.28 

Resistance  of  wings  reduced  to 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

the  mean  density  of  the  air, 

1.338 

2.602 

3.941 

5.183 

6.502 

7.867 

9.166 

10.458 

Coefficient  ~K1  of  the  square  of 

.15818 

.15618 

.15077 

.15355 

.14986 

.15711 

.15494 

.15818 

the  velocity, 
Mean  Kj 

=.1548* 

Velocity  answering  to   the  for-]  yds.  I  yds.   I  yds.      yds.  I  yds.      yds.  i  yds.  i  yds. 
mula,  I  2.918 1  4.129  i  5.108     5.87   !    6.58     7.25   |  7.83   |  8.37 


*  A  review  of  the  coefficient  gives  slight  variations  from  those  recorded 
by  Morin,  the  mean  of  which  would  be  K^.1004  instead  of  .1002. 


RESISTANCE   OF   FLUIDS.  4:19 

This  comparison  of  the  results  of  experiments  with 
those  of  the  above  formula,  show  within  what  limits  of 
exactness  the  latter  represents  the  real  eifects. 

337.  Method  of  reckoning  the  effects  of  acceleration.  — 
We  have  already  shown,  in  No.  327,  that  in  elastic  fluids, 
the  resistance  must  depend  upon  the  acceleration  of  mo- 
tion, and  if  these  considerations  are  admitted,  it  follows 
that  the  resistance  of  the  air  in  variable  motion,  must  be 
represented  by  a  formula  of  the  form  of 


The  experiments  upon  uniform  motion  having  already 
furnished  the   approximate   values  of  K/  and  K,,  it  re- 


mains to  find  that  of  Ka,  or  rather  the  term     2 

t 

Without  going  into  the  details  of  the  calculations,  we 
limit  ourselves  to  pointing  out  the  method  followed,  since 
it  shows  a  remarkable  example  of  the  advantages  to  be 
derived  from  a  graphic  representation  of  the  law  of 
motion. 

In  the  actual  case,  this  law  being  represented  by  a 
continuous  curve,  whose  abscissae  indicate  the  number  of 
turns,  or  the  spaces  described,  and 
whose  ordinates  express  the  times, 
it  is  clear  that  for  one  of  these 
tangents,  MP,  for  example,  the 
ratio  of  NP  to  MN,  in  the  trian- 
gle MNP,  will  be  the  same  as 
that  of  e  to  £,  representing  by  e  j>  E  N 

the   infinitely  small  increase  of 

the  abscissa  in  passing  from  the  point  M  to  the  infinitely 
near  point  M',  and  by  t  the  corresponding  increase  of 

time  or  of  the  ordinate  :  this  ratio  -   of   the    elementary 


420  RESISTANCE   OF  FLUIDS. 

path  to  the  element  of  time  in  which  it  was  described,  is 
precisely  what  is  termed  the  velocity,  which  we  express 

by  the  relation  Y=-,  and  we  see  that  we  may,  by  means 
t 

of  the  graphic  trace  of  Fig.  121,  form  a  table  of  the  simul- 
taneous values  of  the  times  and  velocities,  and  so  construct 
a  new  curve,  whose  abscissae  shall  be  the  times  T,  and 
whose  corresponding  ordinates  shall  be  the  velocities  Y. 

This  new  curve  (Fig.  122) 
yields  to  analogous  considera- 
tions ;  the  tangents,  at  the  differ- 

AB 

ent  points,  give  us  the  ratio  -^^, 

which  is  equal  to  the  acceleration 

c  T   IB         v    ,  .     ,,     , 

FIG.  122.  -> v  being  the  elementary  increase 

t 

of  the  ordinate  or  of  the  velocity  Y,  and  t  being  always 
the  elementary  increase  of  the  time. 

Consequently,  knowing  at  each  instant  the  total  re- 
sistance R,  or  the  portion  of  the  motive  effort  employed 
in  overcoming  the  resistance  of  the  air,  as  well  as  the 

coefficients  K/  and  K15  we  may  calculate  the  term  KaA- 

t 

and  so  deduce  the  value  K2. 

This  process  may  be  abridged,  by  operating  upon  that 
part  of  the  curve  relating  to  the  end  of  the  fall,  since  the 
variations  of  inclination  of  the  tangents  of  the  first  curve 
are  so  small,  that  instead  of  tracing  them,  we  may  deter- 

17 TJV 

mine  them  by  the  value  of  the  quotient  = — ~,  of  the  dif- 
ference of  two  consecutive  spaces  divided  by  that  of  the 
corresponding  times. 

I  dwell  no  longer  upon  this  matter,  and  close  with 
stating  that  this  delicate  and  ingenious  mode  of  discussion 
has  led  M.  Didion  to  assign  to  the  coefficients  of  the  for- 
mula, which  represents  the  law  of  the  resistance  of  air  to 


RESISTANCE   OF  FLUIDS.  4:21 

the  accelerated  motion  of  descent  of  a  plate  1.196  sq.  yds. 
of  surface,  the  following  values : 

K=Olbs.06633+0.1295Y2+0.27652-, 

t 

which  is  reduced  in  case  of  uniform  motion  to 

B=0lbs.06633-}-0.1295Ya, 
for  one  square  yard  of  surface,  Y  being  in  yards. 

338.  Proof  of  the  exactness  of  this  formula. — To  show 
a  posteriori  that  this  formula,  composed  of  three  terms, 
represents  the  law  of  the  resistance  in  accelerated  motion 
more  exactly,  than  those  which  only  contain  a  term  pro- 
portional to  the  square  of  the  velocity,  or  two  terms,  the 
one  constant,  and  the  other  proportional  to  the  square  of 
the  velocity,  M.  Didion  has  first  sought  for  the  values  of 
the  constant  coefficients  which  it  was  proper  to  admit  for 
each  of  these  formulae,  so  as  to  render  them  as  exact  as  pos- 
sible, and,  after  having  found  them,  he  calculated,  by  a 
very  simple  analytical  method,  the  values  of  the  times 
corresponding  to  the  regularly  increasing  spaces  described 
by  the  bodies,  such  as  would  be  furnished  by  these  for- 
mulae, and  he  has  compared  them  with  the  real  times 
furnished  by  the  curve  of  the  law  of  motion.  From  the 
results  of  this  comparison,  which  for  one  particular  case 
are  entered  in  the  following  table,  we  see  that  the  formula 
with  three  terms  of  resistance,  represents,  quite  truly,  the 
law  of  accelerated  motion  of  the  descent  of  a  body  in  air, 
while  the  suppression  of  the  -term  depending  upon  the 

acceleration  -  does  not  admit  of  so  exact  a  representation 
TJ 

of  this  law,  even  in  determining  the  coefficients  so  as  to 
reproduce  the  calculated  duration  for  one  of  the  spaces, 
and  that  is  also  the  case  when  we  suppress  the  constant 
term. 


422 


RESISTANCE   OF   FLUIDS. 


The  only  results  inserted  in  the  table  are  those  of  ex- 
periment No.  6,  during  which  the  temperature  was  at 
62°.24'  (Fah.)  and  the  barometric  pressure  at  2.465  ft,  of 
mercury. 

Comparison  of  times  and  velocities  of  the  fall  of  a  plate 
one  metre  square  =1.196sq-  yds-  observed  and  calculated. 


Spaces  described. 

Observed  dura- 
tions. 

i 
1 

1 

0 

Durations  calculated  by  the  formula 

Velocities  calcula- 
ted by  formula 

0) 

«S   9 

•fi  • 

g  «§ 

yds. 

seconds. 

yds. 

seconds. 

seconds. 

seconds. 

yds. 

0.0999 

0.176 

0.178 

0.160 

0.160 

0.1993 

0.254 

0.253 

0.227 

0.226 

0.2998 

0.306 

0.310 

0.278 

0.277 

0.3994 

0.359 

0.358 

0.322 

0.321 

0.4809 

0.400 

0.400 

0.360 

0.358 

0.5997 

0.428 

0.428 

0.394 

0.393 

0.6996 

0.474 

0.473 

0.419 

0.417 

0.7996 

0.508 

0.506 

0.460 

0.457 

0.8995 

0.537 

0.536 

0.488 

0.487 

0.9995 

0.566 

0.566 

0.518 

0.515 

1.2018 

0.619 

0.622 

0.570 

0.567 

1.3998 

0.679 

0.679 

0.619 

0.617 

1.5988 

0.725 

0.723 

0.665 

0.663 

1.7990 

0.771 

0.771 

0.710 

0.707 

1.9991 

0.815 

0.820 

0.748 

0.746 

3.46 

2.9987 

1.013 

1.013 

0.947 

0.943 

5.46 

3.9919 

1.187 

6.07 

1.186 

1.123 

1.120 

6.05 

4.8098 

1.346 

6.50 

1.346 

1.289 

1.287 

6.49 

5.9974 

1.493 

6.91 

1.497 

1.452 

1.451 

6.86 

6.9970 

1.636 

7.25 

1.639 

1.607 

1.606 

7.14 

7.9966 

1.771 

7.50 

1.776 

1.760 

1.760 

7.37 

8.9962 

1.910 

7.60 

1.912 

1.912 

1.912 

7.56 

9.8133 

2.034 

7.62 

2.042 

2.062 

2.064 

7.69 

339.  Influence  of  the  extent  of  surfaces.  —  To  establish 
this  influence,  M.  Didion  used  a  square  plate  0.5468  yds. 
per  side,  and  so  having  an  area  of  Osq-  yds-299,  or  equal  to 
a  quarter  of  that  of  the  first  plate.  In  calculating  the 
time  of  the  fall  by  the  same  method  as  for  the  plate  of 
1.196  sq.  yds.,  and  by  means  of  the  same  formula 


=     0lbs.06638+0.1295V3  +  0. 


RESISTANCE   OF  FLUIDS. 


423 


he  found  between  the  results  of  observation  and  those  of 
calculation  a  coincidence  quite  sufficient  to  permit  him  to 
conclude,  that  between,  the  extended  limits  in  which  he 
had  operated,  the  resistance  of  the  air  is  proportional  to 
the  extent  of  the  surfaces.  The  temperature  and  barom- 
etric pressure  were  sensibly  the  same  as  in  the  experi- 
ments reported  in  ~No.  338. 


Comparison  of  the  times  and  spaces  described  in  the  fall 
of  a  plate  of  Osq-yds>299  surf  ace,  from  observation  and 
calculation. 


Spaces  described. 

DIJKATION. 

Observed. 

Calculated. 

yds. 

seconds. 

seconds. 

0.0995 

0.174 

0.173 

0.2001 

0.246 

0.242 

0.2996 

0.301 

0.297 

0.4002 

0.356 

0.343 

0.4811 

0.387 

0.384 

0.5993 

0.425 

0.420 

0.6999 

0.460 

0.454 

0.8322 

0.490 

0.485 

0.9000 

0.519 

0.515 

0.9995 

0.547 

0.543 

1.993 

0.775 

0.767 

2.998 

0.951 

0.939 

3.998 

1.102 

1.085 

4.809 

1.240 

1.215 

5.997 

.361 

1.330 

6.997 

.476 

1.412 

7.996 

.586 

1.527 

8.996 

.693 

1.646 

9.546 

.799 

1.738 

340.  Consequence  of  these  results. — "We  see  by  this 
table,  that  the  calculated  times  of  the  falls  are  sensibly 
the  same,  though  a  trifle  less  than  the  observed  times, 
which  shows  that  if  the  coefficient  of  resistance  varies 
with  the  extent  of  surface,  it  tends  to  diminish  with  the 
diminution  of  surface,  rather  than  to  increase,  as  some 


424  RESISTANCE   OF   FLUIDS. 

authors  have  concluded  from  experiments  made  by  obser- 
vation of  the  motion  of  rotation. 

In  recapitulating,  we  may,  without  fear  of  notable 
error,  admit  in  practice,  that  the  resistance  of  the  air  is 
proportional  to  the  extent  of  the  surfaces. 

341.  Experiments  upon  parachutes. — One  of  the  most, 
useful  questions  among  our  researches  upon  the  resistance1 
of  air,  which  our  means  of  observation  enabled  us  to  re- 
solve, was  an  exact  determination  of  the  resistance  expe- 
rienced by  parachutes.  Their  concave  form  causing,  with 
the  same  surface,  a  marked  increase  of  resistance,  it  was 
easy,  in  this  case,  to  obtain  a  uniform  motion  of  descent, 
which  was  indicated  by  the  curve  representing  the  law  of 
motion,  which,  in  this  case,  degenerated  into  a  straight 
line,  whose  inclination  furnished  the  value  of  the  uniform 
velocity. 

The  parachute  employed  was  composed  of  a  frame 
of  whalebones,  disposed  into  four  equidistant  meridian 
planes,  fastened  upon  a  common  rod,  and  strengthened 
by  stays.  This  frame  was  covered  with  taffeta,  strongly 
stretched,  and  it  was  suspended  upon  a  rod,  at  the  lower 
part  of  which  was  attached  the  additional  weights. 

The  exterior  diameter  of  the  parachute  was  1.461  yds. 
measured  perpendicularly  from  the  sides  of  the  polygon, 
and  1.312  yds.  measured  between  the  nearest  points  of  the 
arcs  formed  by  the  rim.  Its  perpendicular  projection  to 
the  direction  of  motion  varied  from  1.433  sq.  yds.  to 
1.444  sq.  yds.  of  surface.  The  versed  sine  of  curvature  of 
this  parachute  was  1.41  ft.  to  the  plane  of  the  ends  of  the 
whalebones. 

A  discussion  of  the  experiments  in  which  the  velocity 
was  uniform  has  shown  that  the  resistance  of  the  air  to 
the  motion  of  this  parachute  could  also  be  represented  by 
an  expression  composed  of  two  terms,  and  that  it  was 
equal  to  1.936  times  that  of  a  plane  of  the  same  surface, 
that  is  to  say,  nearly  double. 


RESISTANCE   OF  FLUIDS.  425 

It  follows,  from  this,  that  it  may  be  expressed  by  the 
formula 


-yd8-  [Olbs.06638+  0. 

S.  (0lbs.1285+0.2507Y2), 

for  units  of  yards  of  surface  and  velocity,  at  the  ordinary 
density  and  temperature  of  the  air. 

342.  Case  where  the  parachute  presents  its  convexity  to 
the  air.  —  In  reversing  the  parachute,  and  causing  it  to 
descend  with  its  convex  surface  downwards,  we  found  a 
much  less  resistance,  and  equal  O.T68  of  that  of  the  plane 
surface  with  the  same  area.     So  that,  in  this  case,  the  re- 
sistance is  represented  by  the  formula 

K=0.768A8q-  ^'  (0lbs.06638+0.1295V2)= 
A(0Ibs.0509+0.0994V2). 

We  see  by  this  that  the  resistance  of  the  same  body 
varies  in  the  ratio  of  1.936  to  0.768,  or  from  2.5  to  1,  ac- 
cording as  it  presents  to  the  air  its  concavity  or  convexity. 

343.  Case  where  the  motion  of  the  parachute  was  ac- 
celerated. —  In  this  expression  of  resistance  we  also  admit 
the  necessity  of  introducing  a  term  dependent  upon  the 

*M 

acceleration  of  motion-,  and  this  expression  for  the  para- 
t 

chute  employed  is 

K=  A  (olbs.1290+0.2513Va+0.2394-), 

in  units  of  yards  for  area  and  velocity. 

A  comparison  of  the  observed  times  of  the  fall  with 
those  deduced  from  this  formula  has  shown  that  it  repre- 
sents the  circumstances  of  motion  with  all  desirable 
accuracy. 


426 


RESISTANCE   OF   FLUIDS. 


344.  Resistance  to  the  motion  of  inclined  planes  in 
air. — These  experiments  were  made  by  means  analogous 
to  those  above  described,  by  causing  to  descend  two 
jointed  planes,  1.0963yds.  long  by  0.5486yds.  wide, 
whose  angles  were  varied  at  intervals  of  5°  from  5°  up  to 
180°,  where  they  form  a  single  plane.  The  results  regu- 
larly observed  from  180°  to  130°  have  shown  that  the 
resistance  decreases  proportionally  with  the  angles,  so 
that,  calling  a  the  angle  of  one  of  the  planes  with  the 
direction  of  motion,  the  resistance  was  expressed  for  uni- 
form motion  by  the  formula 

K=-^A(0lb9.06638-{-.1295V2),  in  units  of  yards. 
90 

A  comparison  of  the  observed  resistance  with  those 
calculated  by  this  formula  show  a  satisfactory  agreement. 


Comparison  between  the  observed  and  calculated  resist- 
ances, for  differently  inclined  planes. 


Angles  formed  by  each  of 
the  planes  with  the  di- 
rection of  motion. 

KESISTANCES 

in  the  ratio  to  those  of  a  plane  perpendicular  to  the  direction 
of  motion. 

-^. 

Observed. 

Calculated. 

90° 
87.5 
82.5 
80. 
77.5 
70. 
67.5 
65. 

1.0000 
0.996 
0.865 
0.856 
0.846 
0.773 
0.737 
0.728 

1.000 
0.972 
0.917 
0.889 
0.861 
0.778 
0.750 
0.722 

We  remark  that  these  results  relate  to  the  case  of  two 
equal  and  jointed  planes,  moved  in  the  air,  with  the  edge 
of  intersection  in  front,  and  are  by  no  means  applicable 
to  the  case  of  isolated  planes. 

The  law  of  the  variation  of  the  resistance  proportion- 


RESISTANCE   OF  FLUIDS.  427 

ally  to  the  angles,  is  also  that  which  we  found  for  water, 
in  operating  upon  cones  of  different  acuteness  (No.  305). 

345.  General  conclusions  from  the  experiments  at 
Metz.  —  In  conclusion,  the  reported  experiments  which 
have  been  made  with  chronometric  mechanism,  giving 
the  times,  to  nearly  some  thousandths  of  seconds,  and  the 
velocities  acquired  at  any  instant  nearly  to  a  hundredth, 
in  observing  the  law  of  descent  in  air  of  different  sized 
plates,  of  two  plates  inclined  towards  each  other,  and 
that  of  a  wheel  with  wings,  for  which  the  velocities 
have  not  exceeded  from  29  to  33  ft.  per  second,  have  con- 
ducted us  to  the  following  conclusions  : 

1st.  In  the  uniform  motion  of  a  body  in  air,  the  resist- 
ance experienced  is  proportional  to  the  extent  of  its  sur- 
face, and  to  another  factor  composed  of  two  terms,  the 
one  constant  and  the  other  proportional  to  the  square  of 
the  velocity. 

As  it  was  easily  foreseen,  that  the  number  of  molecules 
of  the  air  shocked  by  the  displacement  of  the  body  must 
increase  in  the  same  ratio  with  its  density,  the  general 
expression  of  the  resistance  should  contain  a  factor  relative 
to  this  density  ;  so  that  calling  d  the  density  of  the  air  at 
the  temperature  and  pressure  observed,  and  d^  its  density 
at  50°  (Fah.)  and  at  76  centigrades  (or  29.92  in.)  of  barom- 
etric pressure,  and  preserving  t^e  preceding  notations, 
this  resistance  is  represented  by  the  following  formulae  : 

Thin  plates  perpendicular  to  the 

1  Olbs.066  +  0.129V2  I 

I  °  -129  +  °-251V2  ( 

)  °  -051+0.994V2  > 
Two  jointed  plates,  inclined  towards  each 

other  .............................................  R=A|^  j  0  .066  +  0.129V2  I 

The  wings  of  a  fan  wheel  .......................  R=AJ1    }  0.08002-fO.  1545V2  i 


direction  of  motion  ...........................  R=A7 


Parachutes  .........................................  E=Adl 

Parachutes  reversed  ...........  .  ..................  'R=Ad^l 


428  RESISTANCE  OF  FLUIDS. 

It  may  be  observed  that  this  last  formula  accords  in  a 
satisfactory  manner  with  the  results  of  M.  Thibault's  ex- 
periments. 

2d.  In  accelerated  motion  we  must  add  to  the  pre- 
ceding expression  a  term  proportional  to  the  acceleration 
of  motion,  and  the  resistance  is  then  represented  by  the 
following  formulae : 

Thin  plates  perpendicular  to  the 

direction  of  motion E= A^        $  Olbs  .066  + 0.129V2 +  0.276-  V 

Parachutes U=Adl       \  °      >128+ 0.251V2 +  0.2394y  I 

34:6.  The  effort  exerted  ly  the  wind  upon  immovable 
surfaces  opposed  to  its  direction. — We  have  but  few  re- 
sults of  experiments  upon  the  law  and  intensity  of  the 
efforts  exerted  by  the  wind  upon  the  surfaces  exposed  to 
its  action.  Smeaton,  in  his  researches  upon  wind  and 
water,  mentions  a  table  which  was  communicated  to  him 
by  Rouse,  an  English  philosopher.  It  is  reported  in  many 
works,  and  especially  in  Jamieson's  Dictionary  of  Me- 
chanical Science.  Smeaton  says  that  it  was  constructed 
with  great  care  by  Rouse,  after  a  considerable  number  of 
facts  and  experiments.  He  observes  that  for  velocities 
over  50  miles  per  hour,  or  73.33  ft.  per  second,  these 
experiments  do  not  merit  the  same  amount  of  confidence 
as  for  inferior  velocities.  The  comparative  numbers  given 
in  this  table  for  the  efforts  seem  to  have  been  calculated, 
in  admitting  that  the  effort  exerted  is  proportional  to  the 
square  of  the  velocity  of  the  wind,  and  would,  in  general, 
be  represented  by  the  formula 

F=0.00228AV2  in  units  of  feet, 

A  being  the  surface  perpendicular  to  the  action  of  the 
wind ; 

Y  the  velocity  in  feet  per  second ; 

F  the  effort  exerted. 


RESISTANCE   OF   FLUIDS. 


^Efforts  exerted  ly  the  wind  upon  a  surface  one  foot 
square,  placed  perpendicularly  to  its  action. 


Common  designation  of  the 

wind. 

Feet  per 
second. 

Force  exerted 
upon  a  square  foot. 

'    ft. 
1  47 

Ibs. 
005 

< 

2.93 

.020  ) 

| 

4.40 

5.87 

.044  f 
.079  J 

.J 

7.33 
14.66 

.123  J 
.492  i 

Very  brisk  

( 
\ 

22.00 
29.34 

1.107  f 
1.968  i 

36.67 
44.01 

3.075  J 
4.429  i 

51.34 

58.68 

6.027  f 

7.873  ; 

66.01 
73.33 

9.963  j" 
12300 

88.02 

17.715 

11736 

31490 

Hurricane  that  tears  up  trees 

and  over-  ( 

146.66 

49.200 

347.  Observation  upon  the  velocity  of  the  wind. — The 
velocity  of  the  wind  attains,  and  even  sometimes  exceeds 
the  values  above  indicated,  as  is  proved  in  aeronautic 
ascensions.  "We  cite,  among  others,  that  of  Lunardi,  who, 
in  an  ascent  made  at  Edinburgh,  when  the  air  was  cairn 
at  the  surface  of  the  earth,  was,  at  a  certain  height,  borne 
by  a  current  of  air  with  a  velocity  of  TO  miles  per  hour, 
or  102  ft.  per  second ;  that  of  Garnerin,  from  London  to 
Colchester,  in  1802,  where  the  velocity  rose  to  80  miles 
per  hour,  or  117  ft.  per  second ;  and,  finally,  that  of  Green 
in  1823,  who  was  carried  at  the  rate  of  210  ft.  per  second 


*  I  have  copied  this  directly  from  the  English  table.     The  data  in  this 
case  lead  to  the  formula 


instead  of 


F=0.120lAV2  in  killogrammes  and  metres, 
F=0.1163AV2,  as  Morin  has  it. 


430  RESISTANCE   OF   FLUIDS. 

without  accident.  These  velocities  suffice  to  show  what 
difficulties  attend  the  directing  of  balloons.  We  will 
shortly  return  to  this  matter. 

348.  Means  employed  to  measure  the  velocity  of  air. — 
The  difficulty  of  measuring  the  velocity  of  the  air  with 
precision  has  been,  and  still  is,  the  chief  obstacle  against 
the  attainment  of  conclusive  experiments  which  shall 
indicate  the  laws  of  the  effort  exerted. 

The  most  general  mode  adopted  by  experimenters 
consists  in  throwing  to  the  wind  light  bodies,  such  as 
feathers,  thistledown,  the  smoke  of  powders,  or  the  essence 
of  turpentine,  and  in  observing  the  distances  described 
with  the  corresponding  times,  in  the  movement  of  trans- 
lation. But  this  simple  method  affords  but  little  precision 
on  account  of  the  small  distances  in  which  they  can  be 
observed. 

Anemometers,  composed  of  a  small  light  fan-wheel, 
whose  motion  is  transmitted  to  a  counter  which  registers 
the  number  of  turns,  are  most  certain,  and  convenient  for 
use,  though  they  must  previously  be  tested,  or  the  rela- 
tion existing  between  the^  velocity  of  the  wind  and  the 
number  of  turns  of  the  wings  must  be  accurately  deter- 
mined; this  determination  presents  great  difficulties. 

Most  generally,  we  accomplish  this  test  by  placing 
the  instrument  upon  the  horizontal  arm  of  a  species  of 
horse-gin  with  a  vertical  axis,  which  is  made  to  turn  as 
uniformly  as  possible.  We  then  observe  simultaneously 
the  number  of  turns  of  the  wings  and  the  velocity  of 
translation  of  the  instrument,  and  then  suppose  that  the 
effect  produced  by  this  movement  of  the  apparatus  in  the 
air,  the  same  as  that  which  would  be  due  to  the  action  of 
the  wind  impressed  with  the  velocity  of  transport  of  the 
anemometer,  upon  the  wings  of  the  instrument  at  rest.  I 
shall  shortly  point  out  another  process  which  I  have  suc- 
cessfully used  in  great  velocities,  but  first  will  describe  a 


RESISTANCE   OF   FLUIDS. 


431 


very  light  anemometer  which  M.  Combes,  Inspector  Gen- 
eral of  Mines,  constructed  to  measure  the  small  velocities 
of  air,  principally  in  the  ventilation  of  mining  works. 

349.  Anemometer  of  M.  Combes. — We  copy  from  this 
learned  engineer  the  description  which 'he  has  given  in 
the  "  Annales  des  mines"  third  series,  of  the  instrument 
which  he  used  for  the  experiments  in  question. 

"This  instrument  is  similar  to  "Woltiman's  mill  for 
gauging  streams  of  a  considerable  section.  It  is  composed 
of  a  very  delicate  axle,  (turning  in  agate  caps,)  upon  which 
are  mounted  four  plane  wings,  equally  inclined  as  to  a 
plane  perpendicular  to  the  axis.  In  the  middle  of  the 
axle  (figure  123)  is  cut 
an  endless  screw,  which 
drives  a  small  wheel  B, 
with  a  hundred  teeth,  so 
that  the  latter  advances 
one  tooth  for  each  revolu- 
tion of  the  axle  bearing 
the  wings.  The  axle  of 
the  first  wheel  carries  a 
small  cam,  which  acts 
upon  the  teeth  of  a  second 
wheel  E/.  The  last 
held  fast  by  a  claw 


is 


or 


FIG.  123. 


very  flexible  steel  spring,  which  is  attached  to  the  hori- 
zontal plate  upon  which  the  instrument  is  mounted.  At 
each  revolution  of  the  first  wheel  with  a  hundred  teeth, 
driven  by  the  endless  screw,  the  cam  starts  one  tooth  of 
the  second  wheel  with  fifty  teeth ;  the  two  wheels  are 
marked  at  intervals  of  10  teeth.  The  first  from  1  up  to 
10,  the  second  from  1  to  5.  The  index  pointers  fixed  upon 
light  uprights,  which  bear  the  axle  of  the  wings,  serve  to 
mark  the  number  of  teeth  which  each  wheel  has  advanced, 
and  thus  to  indicate  the  number  of  revolutions  of  the  axle 
of  the  wings.  By  means  of  a  detent  and  two  cords,  which 


432  KESISTANCE   OF   FLUIDS. 

move  it,  we  may,  at  a  distance,  arrest  the  rotation  of  the 
wings,  or  allow  them  to  tarn,  under  the  impulse  of  the 
current  of  air  which  strikes  them." 

The  manner  of  using  this  instrument  is  easily  under- 
stood after  this  description.  We  place  the  limbs  at  zero, 
and  the  instrument  in  the  axis  of  the  air  tubes,  keeping 
the  limbs  immovable,  by  means  of  a  catch,  which  is 
loosened  at  the  moment  of  commencing  the  observation, 
and  made  fast  at  the  end  of  the  same. 

It  is  well  to  prolong  the  observation  as  long  a  time  as 
possible,  and  for  two  or  three  minutes  at  least,  if  it  can 
be  done.  The  division  of  the  limbs  does  not  admit  of 
counting  over  5000  turns,  which,  for  a  velocity  of  air 
9.84ft.  per  second  would  only  correspond  with  a  duration 
of  about  2.8  minutes. 

The  test  or  error  of  these  instruments  may  differ  very 
much  from  each  other,  though  their  dimensions  may  seem 
identical  in  all  points.  It  should  then  be  made  for  each 
one  in  particular,  and  repeated,  as  far  as  possible,  when- 
ever we  wish  to  use  it  after  an  interruption. 

Thus  the  anemometer  No.  3,  whose  trial  was  reported 
by  M.  Combes,  gave 

<y=0.8458ft-+0.3005^, 

v  being  the  velocity  of  the  air  in  feet  per  second, 
and  n  the  number  of  turns  of  the  wings  in  V  . 
Another  anemometer  of  the  same  model  gave   the 
relation 


350.  Remarks  upon  the  use  of  the  instrument.  —  This 
little  instrument  is  handy  for  the  measurement  of  small 
velocities,  since  we  see  that  it  can  appreciate  those 
from  .492  to  0.82  ft.  per  second.  In  this  case  it  works 
long  enough  to  give  sufficiently  exact  indications  in  prac- 
tice, still  with  this  condition,  that  the  current  shall  be 


RESISTANCE  OF  FLUIDS.  433 

continuous  and  tolerably  regular,  such  -as  is  the  case  with 
mines  whose  ventilation  is  produced  by  permanent 
causes  slightly  varying  from  one  instant  to  the  other. 

But,  when  accidental  circumstances,  during  the  period 
of  an  experiment,  change  materially  the  velocity  of  the 
current  of  air,  as  happens  in  the  case  of  ventilation  of 
crowded  assemblies,  of  hospitals,  &c.,  by  the  opening  and 
shutting  of  doors,  which  produce  very  great  intermissions, 
we  must  have  an  instrument  to  work  for  a  longer  time,  to 
obtain  more  reliable  mean  results. 

On  the  other  hand,  if  we  wish  to  operate  upon  the 
wind,  whose  intensity  often  varies  very  suddenly  in  very 
short  periods,  it  is  otherwise  necessary  to  have  a  more 
solid  anemometer. 

351.  New  anemometer. — It  is  with  this  aim  that  I 
sought  to  make  another  anemometer,  based  upon  the  same 
principle,  but  capable  of  resisting  winds  of  great  inten- 
sity, and  of  operating  long  enough  to  furnish  sure  mean 
results.  I  also  wished  that  the  indications  of  the  appara- 
tus, whose  general  arrangement  is  represented  in  figure 


FIG.  124 


124,  should  be  free  from  defects  which  might  occasion  a 
sudden  stop  of  the  counter,  and  I  disposed  it  so  that,  the 
instrument  being  put  in  its  place,  and  its  motion  being 
regularly  established,  the  observer  might,  by  a  small  sys- 
tem of  pointers,  marking  points  of  thick  ink  upon  enamel 
dials,  determine  the  instant  when  it  commenced  counting 
the  time,  and  that  when  it  finished. 

The  making  of  this  instrument,  intrusted  to  M.  Bianchi, 


4:34:  RESISTANCE  OF  FLUIDS. 

lias  received  also  from  this  skilful  artist,  improvements 
which  make  its  working  simple  and  easy. 

The  fly  with  wings  is  placed  upon  a  very  delicate  hard 
steel  arbor,  borne  at  its  ends  upon  two  supports,  and  near 
its  middle  upon  a  third  intermediate  support.  The  holes 
are  lined  with  hard  flints.  An  endless  screw,  with  a  sin- 
gle thread,  arranged  upon  the  axle  of  the  fly,  drives  a 
first  wheel  with  100  teeth,  whose  axle  bears  a  double  cup 
pointer,  placed  before  an  enamelled  dial,  divided  into  100 
parts,  and  upon  which  may  be  counted  the  turns  made  by 
the  fly  up  to  100.  Upon  the  axis  of  the  first  wheel  is 
another  endless  screw  driving  a  second  wheel  with  100 
teeth,  whose  axle  bears  also  a  double  cup  pointer,  placed 
before  a  second  dial  divided  into  100  parts,  and  upon 
which,  this  second  pointer  may  mark  the  turns  of  the 
first  wheel,  or  the  hundred  turns  of  the  fly  with  wings  up 
to  10,000.  Finally,  upon  the  axle  of  this  second  wheel  is 
placed  an  "  argot "  which,  at  each  turn  of  this  axle,  drives 
one  tooth  of  a  minute  wheel  of  50  teeth,  which  admits  of 
our  counting  up  to  500,000  turns. 

From  the  experimental  test,  which  will  be  described 
hereafter,  we  easily  conceive  that  this  disposition  allows 
us  to  count,  during  10  minutes,  the  number  of  turns  cor- 
responding to  a  velocity  of  131  ft.  per  second,  and  so  for 
a  much  longer  time  those  corresponding  with  less  veloci- 
ties. 

The  pointing  apparatus,  ingeniously  arranged  by  M. 
Bianchi,  acts  simultaneously  upon  two  double  pointers, 
by  pushing  or  drawing  a  button  placed  at  the  end  of  a 
rod  1.96  ft.  long,  which  bears  the  instrument;  this  allows 
the  observer  to  be  completely  separated  from  the  current 
of  air :  the  transmission  of  motion  is  made  with  equal 
facility,  whatever  may  be  the  direction  given  to  the  box 
holding  the  fly  with  wings  and  its  counter,  which  may 
be  turned  in  different  directions,  according  to  the  will 
of  the  observer. 


RESISTANCE  OF  FLUIDS.  435 

Exchange  wings,  of  various  diameters,  may  be  sub- 
stituted for  each  other,  as  we  wish  to  render  the  instru- 
ment more  or  less  sensible  of  small  velocities  of  air. 

352.  The  testing  of  the  instrument.  —  The  relation  be- 
tween the  number  of  turns  of  the  fly  and  the  velocity  of 
the  air  was  at  first  determined  in  the  usual  way,  but  in 
placing  the  anemometer  at  the  end  of  the  horizontal  arm, 
of  13.12  ft.  length,  of  a  small  horse-gin  established  in  the 
great  church  of  the  Abbey  St.  Martin's,  a  very  simple 
return  motion  allowed  the  observer,  placed  near  the  ver- 
tical shaft  of  the  gin,  to  operate  upon  the  pointers,  when 
the  motion  had  become  regular. 

We  have  thus  observed,  with  two  fly-wheels  with  dif- 
ferent wings,  the  number  of  turns  and  the  velocities  of 
translation  of  many  anemometers  ;  they  were  graphically 
represented  in  taking  the  number  of  turns  for  ordinates 
and  the  velocities  of  translation  for  abscissae,  and  we  ob- 
served that  all  the  points  thus  determined,  were  found  sen- 
sibly in  the  same  straight  line,  which  cut  the  line  of 
abscissas  in  front  of  the  origin,  showing  that  the  relation 
between  the  velocities  and  the  number  of  turns  was  of 
the  form 


a  representing  the  velocity  of  translation  of  the  instru- 
ment or  the  velocity  of  the  air,  beyond  which  only  the 
passive  resistances  of  the  instrument  began  to  be  over- 
come. 

These  first  experiments  gave  for  the  two  flies  of  the 
anemometer  the  following  results  : 

Experiments,  in  number  50,  made  upon  the  first,  which 
had  the  smallest  wings,  with  velocities  of  translation  be- 
tween 5.24  ft.  and  31.16ft,  gave  results  represented  by 
the  formula 


436  RESISTANCE   OF   FLUIDS. 

Another  series  of  40  experiments,  made  with  the  same 
anemometer,  with  larger  wings,  and  so  more  sensible, 
gave  results  represented  by  the  formula 


353.  Remarks  upon  the  mode  of  testing  the  instru- 
ments. —  The  preceding  remarks  are  upon  the  supposition, 
that  the  action  of  the  air  at  rest  upon  a  body  in  motion, 
is  the  same  for  equal  velocities  with  that  of  air  in  motion 
upon  a  body  at  rest.     Without  pretending  actually  to 
dispute  or  admit  the  difference  which  Dubuat  thought 
could  be  deduced  from  his  experiments  upon  the  two 
modes  of  action,  I  will  simply  say,  that,  actually,  the 
difference,  if  it  exists,  must  be  so  small  that  it  may  be 
neglected.     Within  my  knowledge,  there   are  no   other 
modes  of  procedure,  and  the  following  experiments  will, 
I  think,  confirm  the  exactness  of  the  preceding  formulae.* 

354.  Extension  of  the  test  to  great  velocities.  —  The 
velocity  of  translation  impressed  upon  the  anemometer 
could  not  be  made  to  exceed  that  of  about  33  ft.  in  V.     I 
used,  in  order  to  extend  the  test  of  the  instruments  to  that 
of  great  velocities,  the  following  means  :  a  small  ventilator 
of  0.98  ft.  diameter  with  plane  wings  in  the  direction  of 
the  radius  was  mounted  in  a  cylindrical  tube,  (through 
which  it  drove  the  air,)  whose  cross  section,  as  well  as  that 
of  the  junction-pipe  with  the  envelope,  was  equal  to  the 
surface  of  the  vanes.     This  disposition  was  made  to  pro- 
vide against  any  sensible  variation  in  the  velocity  of  the 
air  during  its  passage. 

The  ventilator  was  moved  by  a  small  steam-engine, 
and  by  means  of  pulleys  with  different   diameters,  its 

*  It  may  be  well  to  observe  that  tbe  winged  mills  of  the  same  kind  used 
in  the  gauging  of  water,  have  given  similar  results  to  the  preceding,  and  that 
particularly  the  experiments  of  the  late  M.  Lapointe,  have  shown  that  the 
relation  v=a  -f  bn  holds  good  even  for  variable  velocities. 


RESISTANCE   OF  FLUIDS.  437 

velocities  could  be  changed  from  127'  up  to  2220  turns 
in  V. 

Commencing  at  'first  with  imparting  very  small  veloc- 
ities, we  used  for  measuring  the  velocity  of  the  air  in  the 
tube  the  test  made  with  the  rotating  apparatus  with  a 
vertical  axle,  and  deduced  the  number  of  turns  of  the 
anemometer  and  the  velocity  of  the  air  in  the  tube  up  to 
a  limit  of  from  33  ft.  to  39.36ft.  in  1". 

Comparing,  then,  the  mean  velocities  of  issue  of  the 
air,  with  those  of  the  vanes  of  the  ventilator,  we  found 
between  them  a  constant  ratio,  so  that  the  velocity  of  the 
ventilator  being  v'  and  that  of  the  air  v,  we  had  the  ratio 


and  consequently 


n\ 

— ,=K  or  v=~ 

v 


, 

or  v  '=-  +—n, 
-K.     K. 

which  shows  that  between  these  limits  the  velocity  of  the 
wings  of  the  ventilator  was  proportional  to  that  of  the 
wings  of  the  anemometer. 

This  being  established,  we  gave  greater  speed  to  the 
ventilator,  and  noted  the  number  of  turns  n,  made  by  the 
anemometer  in  1",  and  admitting  that  the  ratio  K  between 
the  velocities  of  the  air,  and  that  of  the  centre  of  the  wings 
of  the  ventilator,  remains  the  same  in  great  velocities  as 
in  small,  we  deduce  the  mean  velocities  of  the  air  when 
it  struck  upon  the  wings. 

Referring  these  numbers  of  turns  as  ordinates,  and  the 
velocities  as  abscissae,  to  the  same  figure  which  had  been 
constructed  for  the  preceding  experiments,  we  found  that 
the  points  so  determined  were  upon  the  prolongation  of 
the  same  straight  line  which  gave  the  relation 


This  coincidence  in  the  results  of  the  two  series  of 


438  RESISTANCE  OF  FLUIDS. 

experiments,  shows  the  simultaneous  permanence  of  the 
two  relations 

and  v=Kv' 


even  for  great  velocities. 

In  fact,  since  in  taking  for  v  the  values  of  KJy',  we 
have  admitted  for  great  velocities  the  exactness  of  the 
relation 


as  shown  by  the  trace,  it  follows  that  the  ratio^  and  -== 

are  constant,  which  can  only  happen  when  #,  &,  and  K 
are  also  constant. 

Making  ^=c  and  =  -=c',  c  and  c'  being  two  constant 
numbers,  we  deduce 

ir      Cb        j    b       1)G        ,         ,  5      C 

K=-  and  •===—  =c,  whence  -=-.: 
c         K     a  a    c' 

which  necessarily  implies  the  constancy  of  the  number  £>, 
since  the  coefficient  a  is  independent  of  the  velocity  or 
the  number  of  turns. 

Tt  follows  from  these  experiments  : 

1st.  The  observations  made  with  the  ventilator  have 
extended  the  test  of  the  anemometer  with  small  wings  up 
to  velocities  of  about  131ft.,  which  exceeds  the  usual 
wants  of  experiments. 

2d.  That  there  exists  a  constant  ratio  between  the 
velocity  of  rotation  of  ventilators  and  that  of  the  air  which 
they  drive  in,  or  exhaust  from  the  tube. 

This  ratio,  moreover,  depends  upon  the  dimensions 
of  the  pipes,  and  also  upon  those  of  the  central  openings 
of  admission  into  the  ventilator. 

3d.  That  in  future,  and  when  the  ratio  of  the  velocity 
of  the  air  expelled  by  a  ventilator,  given  by  the  number 
of  turns  of  its  wings,  is  known,  we  may  easily  adjust  ane- 


RESISTANCE   OF   FLUIDS.  4:39 

mometers  of  different  kinds,  whether  they  are  those  giv- 
ing the  velocity  by  the  number  of  turns  of  their  wings,  or 
whether  they  are  pressure  anemometers,  which  will  be 
far  more  convenient  than  the  first  means  which  we  used, 
and  admit  of  an  extension  of  the  test  to  great  velocities. 

355.  Experiments  of  M.  Thibault  upon  the  effort  ex- 
erted by  the  wind  upon  immovable  surfaces  exposed  to  its 
action,  perpendicular  to  its  direction.  —  We  are  indebted 
also  to  M.  Thibault  for  some  experiments  which  he  under- 
took as  initiatory  to  some  researches  upon  the  action  of 
the  wind  upon  sails,  in  which  he  used  ingenious  devices 
to  measure  the  efforts  of  the  wind  upon  surfaces  of  a  given 
extent  ;  he  used  an  anemometer  provided  with  a  dyna- 
mometer, and  determined  the  velocity  of  the  wind  in 
giving  to  it  light  feathers  or  thistle-tufts,  and  observing 
the  time  of  their  passage  over  a  given  space.  This  mode 
is  not  exact,  and  may  occasion  some  errors  of  a  nature 
tending  to  influence  the  final  results  of  experiments. 

Admitting,  conformably  to  the  experiments  at  Metz, 
of  which  an  account  has  been  given  in  Nos.  336  and  339, 
that  the  resistance  may  be  expressed  by  the  formula 


in  which  K/=Olbs.068  will  be  the  coefficient  of  the  con- 
stant term,  we  find  that  the  experiments  of  M.  Thibault 
conduct  to  the  following  results,  which  give  us  a  general 
mean  of  the  values  of  the  coefficient  K1} 

1^=0.16601  for  units  of  yards. 


EESISTANCE   OF  FLUIDS. 


Experiments  of  M.  Thibault  upon  the  action  of  wind  upon 
plane  surf  aces  perpendicular  to  its  direction. 


•2 

1 

13 

. 

1 

•d 
a 

i 

I 

i 

* 

FT* 

1 

| 

| 

i 

1 

fj 

B 

Ci 

5o^ 

M  • 

a 

B 

•s 

•33 

.• 

*S- 

«2'o<i 

*s^  i  « 

I*"5 

«g 

i 

-b 

2 

«tf 

S3 

|1w 

0)  §"0^ 

Jt 

a 

o 

I 

3 

•3 

§ 

«  'o 

11*0 

ll> 

£_| 

o 

A 

f 

>• 

H 

0 

1 

'i 

O  ® 

W 

2 

M 

M 

sq.  yds. 

ft 

yds. 

Ibs. 

Ibs. 

Ibs. 

2.512 
2.463 

66.2° 
64.4° 

4.568 
5.308 

0.4907 
0.55048 

0.4821 
0.5419 

0.019316 
0.016077 

0.17886 
0.14886 

0.1291 

2.446 

59° 

5.419 

0.56835 

0.008601 

0.5597 

0.015937 

0.14756 

2.430 

59° 

6.124 

0.99004 

0.9814 

0.021877 

0.20257 

2.466 

58.1° 

8.988 

0.20308 

0.1945 

0.020926 

0.19376 

Mean 

0.17432 

0.1701 

2.449 
2.413 

58.1° 
48°92 

4.651 
2.000 

0.98827 
0.22511 

0.011564 

0.9767 
0.2136 

0.03757 
0.03949 

0.18476 
0.19418 

Mean 

0.18947 

It  follows  that  the  mean  value  of  the  action  of  the 
wind  upon  plane  surfaces,  perpendicular  to  its  direction 
would  be  expressed  by  the  formula 

E= A  (0.068+0.18189V2), 

for  units  of  yards,  if  no  account  is  made  of  the  variation 
of  density  corresponding  to  the  barometric  pressure  and 
to  the  temperature,  which,  in  common  applications,  is  but 
of  little  importance  though  easily  made. 

356.  Agreement  of  these  results  with  those  of  Profes- 
sor Rouse,  cited  ly  Smeaton. — We  would  remark  that, 
with  the  exception  of  the  constant  term  0.068A,  which, 
for  mean  velocities  of  the  wind,  has  a  very  small  influ- 
ence, the  preceding  formula  gives  for  the  resistance  nearly 
the  same  value  as  that  of  No.  346,  representing  the  ex- 


RESISTANCE   OF  FLUIDS.  441 

periments  of  Rouse,  which  seem  to  have  been  made  with 
velocities  far  superior  to  those  observed  by  M.  Thibault. 

Either  formula  may  therefore  be  confidently  employed 
for  great  velocities. 

357.  Observation. — The  experiments  of  Metz  having 
given  for  the  coefficient  the  value  1^= 0.1295,  when  the 
body  moves  in  the  air  at  rest,  it  would  follow,  according 
to  the  ideas  of  Dubuat,  that  the  effort  exerted  by  the  air 
in  motion  upon  a  body  at  rest,  would  be  to  the  resistance 
experienced  by  the  same  body  in  motion  in  the  air,  with 
equal  velocities,  nearly  in  the  ratio  of 

0.1819  to  0.1295  or  of  1.40  :  1. 

358.  Influence  of  the  curvature  of  surfaces. — M.  Thi- 
bault has  made  a  comparison  of  efforts  exerted  by  the 
wind  upon  a  plane  surface,  and  upon  a  canvass  of  double 
curvature,  0.1302  sq.  yds.  of  total  surface,  and,  in  the  last 
case,  capable  of  taking  a  curvature  whose  last  elements 
make,  with  the  direction  of  the  wind,  an  angle  of  from 
50  to  55  degrees.     He  has  found  that,  at  the  same  day, 
and  under  the  action  of  the  same  wind,  the  effort  exerted 
upon  the  plane  surface  was  to  that  exerted  upon  the 
curved,  in  the  ratio  of  0.1079  to  0.1135  or  of  0.951  to  1, 
which  shows  the  difference  to  be  very  slight. 

359.  Influence  of  the  inclination  of  surf  aces  towards 
the  wind. — In  presenting  successively  to  the  action  of  the 
wind,  surfaces  perpendicular  or  oblique  to  its  direction, 
the  author  has  established  the  fact,  that  the  effort  exerted 
upon  a  given  surface  was  not  influenced  by  their  inclina- 
tion, except  wlien  the  latter  attained  the  angle  of  from 
45°  to  50°  with  the  direction  of  the  wind.     "We  may  re- 
member that  a  similar  result  was  obtained  in  the  case  of 
surfaces  moving  in  the  air  at  rest. 


442  RESISTANCE   OF   FLUIDS. 

Other  experiments  of  M.  Thibault  were  made  relative 
to  a  comparison  of  the  velocities  of  wind  and  of  a  vessel 
under  sail,  which  were  but  initiatory  to  those  which  he 
proposed  undertaking  upon  this  important  subject,  when 
a  fatal  accident  deprived  the  navy  of  this  young  and  ac- 
complished officer. 

360.  Difficulties  in  the  directing  of  balloons. — The  fre- 
quent balloon  ascensions  made  within  the  past  few  years, 
have  stimulated  a  great  number  of  attempts  to  direct 
them  in  calm  air,  and  even  against  the  wind,  and  it  may 
not  be  useless  to  say  a  few  words,  serving  to  show  the 
difficulties  of  this  problem,  and  even  the  impossibility  of 
its  solution  with  the  mechanical  means  at  our  disposal. 

We  would  first  remark  that  observation  proves  that  a 
calm  at  the  surface  of  the  earth  is  by  no  means  a  guarantee 
that  the  same  repose  exists  in  the  upper  strata,  even  at 
small  heights,  and  that  consequently  an  apparatus  suffi- 
cient for  a  calm  is  by  no  means  so  for  all  heights. 

General  Meusnier,  of  the  military  engineers,  who  has 
devoted  much  time  to  the  subject  of  balloons,  has  left  a 
memoir  upon  the  subject,  a  succinct  analysis  of  which 
may  be  found  in  the  journal  "  le  Conservatoire,"  No.  1  of 
the  second  year.  We  see  in  this  memoir  that  this  learned 
officer  has  already  presented  the  difficulties  of  the  prob- 
lem in  these  terms : 

"  We  have  examined  the  possible  effects  of  many  of 
the  machines  proposed  for  the  direction  of  balloons :  these 
machines  must  be  moved  by  men  whose  weight  is  great 
compared  to  their  force ;  it  follows  that  they  will  have 
but  little  effect  in  overcoming  the  resistances  of  the  air, 
on  account  of  the  great  surface  of  the  balloons. 

u  Calculation  applied  to  the  means  of  direction,  of 
whatever  character  they  may  be,  shows  in  general,  that 
they  can  never  afford  for  balloons  a  velocity  over  a  league 
(3.64  ft.  per  V)  an  hour  independently  of  the  wind" 


RESISTANCE   OF   FLUIDS.  443 

Colonel  Didion,  in  a  paper  read  before  the  Scientific 
Congress  at  Metz,  in  1838,  has  shown,  by  very  simple 
calculations,  that  the  velocity  impressed  in  calm  weather, 
cannot  exceed  this  limit,  even  in  the  most  favorable  hy- 
potheses, as  to  the  weight  of  men  borne,  of  balloons  and 
their  rigging. 

A  cubic  foot  of  air  at  the  temperature  of  32°  (Fah.) 
and  at  a  pressure  of  2.492ft.  of  mercury,  weighs  0.08118 
pounds,  while  the  same  volume  of  hydrogen  gas,  impure 
and  moist,  such  as  is  made  for  general  use,  weighs 
0.000624  Ibs.  The  difference,  0.0805  Ibs.,  is  the  weight 
which  a  cubic  foot  of  this  gas  can  sustain  in  air.  But  as 
the  air  and  gas  are  in  elevated  regions  subject  to  a  less 
pressure,  they  are  then  dilated,  and  the  volume  which 
the  same  weight  of  the  gas  occupies  will  be  greater,  and 
it  must  be  so  also  with  that  of  the  balloon. 

If  we  admit,  that  to  pass  above  ordinary  moun- 
tains we  must  rise  2624  ft.  above  the  level  of  the  sea,  and 
that  then  the  pressure  will  not  be  over  T%-  of  that  at  the 
surface  of  the  earth,  and  if  the  temperature  is  at  50°,  it 
would  follow  that  the  weight  of  0.000624  Ibs.  of  hydrogen, 
instead  of  occupying  1  cubic  foot,  would  have  a  volume 
of  1.15  cu.  ft.,  and  one  cubic  foot  of  this  gas  will  only 
weigh  0.005433  Ibs.  On  the  other  hand,  the  cubic  foot 
of  air,  whose  pressure  is  T9¥,  with  a  temperature  £=50°, 
will  weigh 


Consequently,  1  cubic  foot  of  the  gas  of  the  balloon 
can  only  be  in  equilibrium  with  a  weight  of 

O.OT0486lb8-—  0.005433  =  0.0650531  bs- 

If  we  admit  that  the  weight  of  a  man  is  not  over 
143.36  Ibs.,  and  that  of  his  skiff  is  but  11.02  Ibs.,  without 
supplies,  the  total  weight  to  be  raised  per  man  will  be 


444  RESISTANCE   OF  FLUIDS. 

154.38  Ibs.,  and  the  balloon  should  have  a  volume  of 
154.38 


0.065053 


=2373™- ft-  per  man  to  be  raised, 


which  corresponds  to  a  sphere  of  16.53  ft.  for  each  man. 

In  taking  account  of  the  weight  of  the  covering,  which 
cannot  be  less  than  0.05122  Ibs.  per  square  foot,  we  find 
that  the  diameter  should  be  18.34  ft. 

Calculating  from  this  basis  the  minimum  diameters 
to  be  given  to  balloons  designed  to  carry  different  num- 
bers of  men,  M.  Didion  found  as  follows  : 

Number  of  men  .........      1        2        34567         89       10 

Weight  to  be  raised...  154.41b.  308.7  463  617.5  772  926  1080  1235  1389  1544 

ft.          ft.      ft      ft.     ft.      ft.      ft.     ft.       ft.       ft 

Diameter  of  balloons...  18.3      22.5  25.4  27.8  29.9  31.8  33.3  34.7  36.1  37.1 

From  known  experiments  the  resistance  of  the  air  to 
the  motion  of  spherical  bodies,  for  velocities  within  3  and 
32  ft.  is  approximately  represented  by  the  formula 


If,  for  example,  there  is  a  balloon  designed  for  one 
man,  we  have 

TV         1  8  34.2 

D=18.34  ft.  and  ==  264.22  •*  ft, 


E=0.1886V2, 

which  gives  the  following  resistances  and  quantities  of 
work  for  different  velocities  : 

Velocities  in  feet  per  second  ......       3.28        6.56        9.84       13.12       16.4 

Resistances  in  pounds  ..............       2.030      8.125  18.267    32.476    50.746 

Ibs.  ft. 

Work  expended  in  1  second  ......       6.659    53.227  179.81     426.19     832.49 

Now,  a  man  in  a  day's  work  of  8  hours  cannot,  under 
the  most  favorable  circumstances,  and  with  the  most  ap- 


RESISTANCE   OF  FLUIDS.  445 

proved  mechanism,  develop  a  work  over  from  43  Ibs.  ft. 
to  58  Ibs.  ft.  in  1". 

"We  see,  then,  even  admitting  that  there  is  no  loss  of 
work  arising  from  the  passive  resistances  of  the  machinery, 
which  can  never  be,  the  greatest  velocity  that  a  man  can 
impart  to  his  balloon  is  6.56  ft.  per  second,  or  4.4  miles 
per  hour  in  a  calm. 

As  for  other  motors,  such  as  steam-engines,  their  own 
weight,  that  of  the  fuel,  and  of  the  water,  would  tend  to  give 
such  dimensions  to  the  balloon,  that  the  work  of  the  resist- 
ance of  the  air  in  small  velocities,  would  prevail  over 
that  which  could  be  developed  by  the  motive  apparatus. 

Finally,  in  the  present  state  of  our  knowledge,  and 
progress  in  the  mechanical  arts,  the  solution  of  the  ques- 
tion of  aerial  navigation  is  shut  in  within  a  circle  beyond 
which  we  cannot  pass,  without  discovering  some  new 
motor  at  once  light  and  powerful,  in  relation  to  the  quan- 
tity of  work  to  be  developed. 


THE  END. 


FIG.  16'. 

Longitudinal  Elevation  of  Figure  16. 


See  page  37,  Article  40. 


FIG.  20'. 


FIG.  20". 

The  counter. 


Art.  46,  page  46. 


Side  Elevation — Rotating  dynamometer  with  counter — Scale  Vao  (Article  53). 


FIG.  22'. 


FIG.  22". 

Rotating  dynamometer  with  counter — Article  on  CD— Scale  Van. 


FlG.    86'. 
Apparatus  for  tabulating  curves — Scale  Ye 


See  Article  79,  page  88. 


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days  prior  to  due  date. 

DUE  AS  STAMPED  BELOW 

MAR  0  9  W98 


12,000(11/95) 


